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Integrating stand and forest models for decision analysis Williams, Douglas Harold 1976

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INTEGRATING STAND AND FOREST MODELS FOR DECISION ANALYSIS by DOUGLAS HAROLD WILLIAMS B . S c , Simon Fr a s e r U n i v e r s i t y , 1970 M.Sc, U n i v e r s i t y o f B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the FACULTY OF GRADUATE STUDIES Department of F o r e s t r y We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1976 CcT) Douglas Harold Williams In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the l i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of the thesis for s c h o larly purposes may be granted by the Head of my Department or by h i s representatives. I t . i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Forestry The University of B r i t i s h Columbia 2075 Westbrook Place Vancouver, Sanada V6T 1W5 A p r i l 29, 1977 i i ABSTRACT SUPEHVISOH: A. KOZAK Models of f o r e s t stands and management u n i t s have gained wide acceptance as t o o l s f o r planning f o r e s t lands management. T h i s t h e s i s i n t e g r a t e s t h e i r use i n a framework amenable to d e c i s i o n a n a l y s i s . The f o r e s t lands planning process of the B r i t i s h Columbia F o r e s t S e r v i c e i n v o l v e s a m u l t i l e v e l h i e r a r c h i c a l s t r u c t u r e . At each l e v e l , a planning problem was i d e n t i f i e d and c h a r a c t e r i z e d by one cr more management o b j e c t i v e s and a set of management d e c i s i o n s , seme of which are c o n s t r a i n e d by neighbouring l e v e l s of the planning c h a i n . At any l e v e l , a model of the d e c i s i o n problem should be capable of l i n k i n g t o the other l e v e l s -of the planning process. The c h a i n should not be ' s e p a r a t e d 1 and optimized at each l e v e l . M u l t i s t a g e a n a l y s i s was used to examine the u n d e r l y i n g mathematical s t r u c t u r e of the combined management u n i t and stand d e c i s i o n problem, i d e n t i f y i n g the d e c i s i o n and s t a t e v a r i a b l e s , and o b j e c t i v e and c o n s t r a i n t f u n c t i o n s . The components of the combined problem were i d e n t i f i e d as s c h e d u l i n g management a c t i o n s on a stand or treatment u n i t , and a l l o c a t i n g c o n s t r a i n e d commodities a c r o s s a management u n i t . S e v e r a l approaches to o p t i m i z a t i o n of the two l e v e l problem were considered. Scheduling management a c t i o n s on a treatment u n i t i s best accomplished through a l g o r i t h m s t h a t e x p l o i t the s e r i a l m u l t i s t a g e s t r u c t u r e of the problem. However, the commodity a l l o c a t i o n component cannot be o p t i m i z e d v i a the usual dynamic programming r e c u r s i o n s , due to a continuous and i i i m u l t i d i m e n s i o n a l s t a t e space. The d i s c r e t e maximum approach improves computational e f f i c i e n c y with regard to the s t a t e space but, as with dynamic programming, i t r e g u i r e s d e c i s i o n i n v e r s i o n of the t r a n s i t i o n f u n c t i o n s to handle boundary values of the commodity s t a t e s . C o n v e r s e l y , approximation of the commodity a l l o c a t i o n problem with a l i n e a r model and o p t i m i z a t i o n by l i n e a r programming i s very e f f i c i e n t , but t r e a t s the stand l e v e l subproblem inadequately. fitter c o n s i d e r a t i o n of the r o l e of Lagrange m u l t i p l i e r s i n the d i s c r e t e optimum f o r m u l a t i o n of the problem, the l i n e a r and m u l t i s t a g e approaches were s y n t h e s i z e d through Dantzig-Wolfe decomposition. The g e n e r a l problem of f i n d i n g an o p t i m a l sequence of management a c t i o n s with a stand s i m u l a t i o n model was examined and two technigues c o n s i d e r e d . F i r s t , a c o n v e r s a t i o n a l s u p e r v i s o r system was used i n t e r a c t i v e l y to explore the o b j e c t i v e s u r f a c e of a model. The optimum was sought by a p a t t e r n search technigue, the s e q u e n t i a l simplex a l g o r i t h m . Second, the stand model was embedded i n a network f o r m u l a t i o n t h a t vas optimized through dynamic programming. Both techniques were t e s t e d with s i n g l e t r e e / d i s t a n c e independent and whole stand models. The network f o r m u l a t i o n of the stand model was combined with a l i n e a r management u n i t model (Timber RAM) and optimized by Dantzig-Wclfe decomposition. The decomposition system was demonstrated with a management u n i t of 85,000 ha. s i m u l a t e d by 8 submodels, under c o n v e n t i o n a l present net worth and mixed g o a l o b j e c t i v e s . i v The decomposition system combines three powerful components: an e x i s t i n g , o p e r a t i o n a l model of management u n i t planning i s optimized by a commercially a v a i l a b l e mathematical programming system, with f o r e s t stand models p r o v i d i n g a c c u r a t e and d e t a i l e d estimates of responses to management a c t i o n s . The decomposition approach was economically a t t r a c t i v e , s o l v i n g problems of f a r g r e a t e r complexity than could be attempted under c o n v e n t i o n a l l i n e a r programming f o r m u l a t i o n s . V TAELE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS v LIST OF FIGURES , . v i i i LIST CF TABLES X ACKNOWLEDGEMENTS x i 1. I n t r o d u c t i o n 1 1.1 Models In The Planning Process ..................... 2 1.2 The Planning Process Of The B r i t i s h Columbia F o r e s t S e r v i c e 3 2. A Review Of F o r e s t Stand And Management Onit Models .... 9 2.1 Stand S i m u l a t i o n Models ............................ 9 2.2 Stand O p t i m i z a t i o n Models 15 2.3 Management Unit S i m u l a t i o n Models .................. 17 2.4 Management Unit O p t i m i z a t i o n Models ................ 20 2.5 Summary ............................................ 23 3. Problem A n a l y s i s - The Process Of A b s t r a c t i o n .......... 24 3.1 Terminology And Notation 24 3.1.1 Land Units 25 3.1.2 Management A c t i v i t i e s 26 3.1.3 State And State T r a n s i t i o n 27 3.1.4 Management O b j e c t i v e s And Returns ............. . 29 3.1.5 Management C o n s t r a i n t s ........................ 30 3.2 General Problem Formulation ........................ 32 3.3 M u l t i s t a g e A n a l y s i s 35 3.3.1 MP 1 As A 2-point Boundary Value Problem ....... 36 v i 3.3.2 MP2 As An I n i t i a l Value Problem 41 3.3.3 MP2 As A Subproblem Of MP1 45 3.3.4 The S e r i a l M u l t i s t a g e Model , 48 3.4 Approaches To O p t i n i z a t i c n 49 3.4.1 Decomposition Ey Dynamic Programming .......... 49 3.4.2 The D i s c r e t e Optimum E r i n c i p l e 54 3.4.3 Approximation Hith A L i n e a r Model 59 3.4.4 Dantzig-Wolfe Decomposition .. 63 4. O p t i m i z a t i o n Of The Subproblem ......................... 67 4.1 D i r e c t O p t i m i z a t i o n ................................ 67 4.1.1 The S e q u e n t i a l Simplex Algorithm .............. 70 4.1.2 The O p t i m i z a t i o n S u p e r v i s o r Program .. 74 4.1.3 Test Case: Meyer's Model 79 4.1.4 Tes t Case: Goulding's Model ., 93 4.1.5 Test Case: K i l k k i ' s Model 104 4.1.6 Summary And D i s c u s s i o n 111 4.2 MP2 As A Network Problem 114 4.2.1 Generating The Graph Of A l t e r n a t i v e Management Seguences ........................ 122 4.2.2 F i n d i n g The Optimum Management Seguence 129 4.2.3 L i m i t a t i o n s Of The D i s c r e t e Formulation ....... 135 4.2.4 Summary And D i s c u s s i o n 139 5. J o i n t O p t i m i z a t i o n Of 8E1 And MP2 Via Decomposition .... 140 5.1 The L i n e a r Program Master Problem 140 5.2 Decomposition Of The L i n e a r Model 143 5.2.1 Dantzig-Wolfe Decomposition With MPSX . 1 4 5 5.3 An Example Problem To Demonstrate The Decomposition Approach i ,t 147 5.3.1 Problem D e s c r i p t i o n 148 5.3.2 The Unconstrained Optimal S o l u t i o n 150 5.3.3 St a r t - u p Procedure And F i r s t F e a s i b l e S o l u t i o n 156 5.3.4 F i n a l S o l u t i o n 157 5.4 A Goal Parametric A n a l y s i s With The Decomposition Model .............................. 164 5.5 Summary And D i s c u s s i o n 167 6. Con c l u s i o n s ., 171 v i i BIBLIOGRAPHY ...................................... 176 APPENDIX 1 - A Summary Of Symbols And Notation ......... 183 APPENDIX II - I t e r a t i v e Approximation Of F i n a l C o n d i t i o n s ................................ 185 APPENDIX I I I - D i s t i n g u i s h i n g P o i n t s On A S t o c h a s t i c Surface 189 APPENDIX IV - L i s t i n g Of SIMOPT Scanner .191 APPENDIX V - Simulated Managed Stand Y i e l d Tables -Meyers ... 198 APPENDIX VI - Simulated Stand Tables - Goulding ......... 203 APPENDIX VII - A M u l t i p l e D e c i s i o n T hinning Problem As A Network Problem ( K i l k k i ' s Scots Pine Model) 206 APPENDIX VIII - A M u l t i p l e D e c i s i o n Thinning Problem As A Network Problem (Goulding's Douglas F i r Model). 218 APPENDIX IX - A wolfe-Dantzig Decomposition Program In MPSX 222 APPENDIX X - Treatment u n i t Management O u t l i n e s For Decomposition Demonstration Problem ....... 226 v i i i LIST OF FIGURES 1. L e v e l s of f o r e s t planning . ........................ 4 2. A con c e p t u a l model of the BCFS pl a n n i n g process ........ 6 3. MP 1 as a 2-point boundary value problem 40 4. MP2 as an i n i t i a l value problem 44 .5. MP2 as a subproblem of M.P1 47 6. Elements of a s i m p l i c i a l s e a r c h : r e f l e c t i o n , expansion, and c o n t r a c t i o n 72 7. The s i m u l a t i o n c p t m i z a t i o n s u p e r v i s o r system ........... 75 8. S e n s i t i v i t y of meyers' model to ranging of the i n t e n s i t y of the f i r s t t h i n n i n g . 89 9. The pathway of the o p t i m i z a t i o n a l g o r i t h m s on a s u r f a c e of present net worth ( K i l k k i ' s model). 107 10. Volume/age curves corresponding to s i x l e v e l s o f d e n s i t y at age 20 years, of Sects pine ( K i l k k i ' s model). .......... ... ..... 115 11. A l t e r n a t i v e management sequences represented as a c y c l i c d i r e c t e d graph. 116 12. A l t e r n a t i v e management sequences represented as an a c y c l i c d i r e c t e d graph of s t a t e s and time stages. ...... 117 13. L a b e l l i n g of arcs produced by the graph g e n e r a t i n g program. 124 14. S t a t e ' d e f i n i t i o n and precedence graph of a m u l t i p l e t h i n n i n g d e c i s i o n problem ( K i l k k i * s Scots pine model) .. 125 15. A management o u t l i n e r e port f o r m u l t i p l e t h i n n i n g s of / Scots pine. ............................................ 127 16. A graph r e p r e s e n t i n g the a l t e r n a t i v e management sequences of a m u l t i p l e t h i n n i n g problem i n Scots pine. 129 . 17. An o p t i m a l management seguence c a l c u l a t e d by embedding K i l k k i ' s Scots pine model i n a network f o r m u l a t i o n ....... 131 18. S e n s i t i v i t y of the o p t i m a l management-sequence to d i f f e r e n t d i s c o u n t r a t e s ( K i l k k i ' s model i n a net-work formulation) 134 < 19. Goulding's model i n a network d e c i s i o n problem, with a s i m p l i f i e d d e f i n i t i o n of the s t a t e . .................... 137 i x 20. Flowchart of the MPSX Dantzig-Wolfe decomposition program ................................................ 146 21. Unconstrained o p t i m a l p o l i c i e s f o r treatment u n i t s 1-4 151 22. Unconstrained optimal p o l i c i e s f o r treatment u n i t s 5-8 152 23. Volume flow graphs f o r three management u n i t plans .... 155 24. Stopping r u l e - a s y m t c t i c behavior of the decomposition model o b j e c t i v e f u n c t i o n ............................... 158 25. C o n s t r a i n e d optimal management sequences treatment u n i t s 1,2,4 and 5 161 26. C o n s t r a i n e d o p t i m a l management sequences treatment u n i t s 6, 7 and 8 162 27. Decadal volume h a r v e s t s computed by g o a l - p a r a m e t r i c programming ............................................ 167 28. ftrea of u n i t s u b j e c t to the management sequences c r e a t e d at each decomposition 169 29. Searching f o r complementary s l a c k n e s s . ....-v.*. 188 30. Precedence graph f o r k i l k k i ' s demonstration problem. .. 216 X LIST OF TABLES 1, Cost and b e n e f i t assumptions used i n the o p t i m i z a t i o n a n a l y s i s of Meyers' model. 81 2. Cost and b e n e f i t assumptions used i n the e v a l u a t i o n of Goulding's model ( a f t e r Hoyer (1975), page 10) ......... 95 •3. S e n s i t i v i t y of the o p t i m a l two t h i n n i n g management seguence to v a r i a t i o n s i n the i n t e n s i t y of the f i r s t t h i n n i n g . ( K i l k k i ' s model) ............ 109 4. S e n s i t i v i t y of the optimal 2 t h i n n i n g management seguence to v a r i a t i o n i n the age of the f i r s t t h i n n i n g . ( K i l k k i ' s model) 110 5. Optimum management seguences with two commercial t h i n n i n g s , at i n c r e a s i n g d i s c o u n t r a t e s . ( K i l k k i ' s model) 111 6. Computer c o s t s f o r the demonstration problem of D a n t z i g - H c l f e decomposition 169 x i ACKNOWLEDGEMENTS The author wishes to thank h i s s u p e r v i s o r . Dr. A. Kozak, f o r h i s guidance and p a t i e n c e throughout the author's p e r i o d of graduate s t u d i e s . Also, the author thanks Mr. G.G. Young f o r i n t r o d u c i n g him to the technigues of o p e r a t i o n s r e s e a r c h and encouraging him i n the w r i t i n g of t h i s d i s s e r t a t i o n . Dr. J.H.G. Smith, Dr. A. Chambers, and Dr. L.G. Mitten reviewed the t h e s i s and provided welcome advi c e and c r i t i c i s m . Dr. J.L. C l u t t e r , who.read the t h e s i s as e x t e r n a l examiner, i s g r a t e f u l l y acknowledged. F i n a l l y , the author would l i k e to express h i s g r a t i t u d e to h i s f r i e n d s and aguaintances at OBC and i n the B r i t i s h Columbia F o r e s t S e r v i c e , who have attempted to give him a r e a l i s t i c p e r s p e c t i v e of f o r e s t r y i n B r i t i s h Columbia. 1 1. I n t r o d u c t i o n D e c i s i o n a n a l y s i s i s the a n a l y t i c a l process by which one s e l e c t s s p e c i f i c courses of a c t i o n from a s e t of p o s s i b l e c ourses of a c t i o n , i n order to achieve h i s g o a l s 1 . In the l a s t f i v e years the a p p l i c a t i o n s of o p e r a t i o n s r e s e a r c h t o f o r e s t l a n d s planning have undergone a s u b t l e r e d e f i n i t i o n , from ' t o o l s f o r d e c i s i o n making' to ' t o o l s f o r a n a l y s i s ' . T h i s s h i f t r e f l e c t s the r e c o g n i t i o n of the l i m i t a t i o n s o f the technology and the r e a l i t y that only the resource manager can make a c r e d i b l e d e c i s i o n . Another aspect of t h i s s h i f t i s the reduced emphasis on developing new techniques and approaches. The next f i v e years w i l l see i n c r e a s i n g emphasis on e f f e c t i v e l y u s i n g the e x i s t i n g t o o l s i n planning systems. This t h e s i s presents some techniques f o r i n t e g r a t i n g e x i s t i n g t o o l s at the management u n i t and s i n g l e stand l e v e l , and e x p l o r i n g planning problems i n a d e c i s i o n a n a l y s i s framework. 1 T h i s d e f i n i t i o n i s provided by S.M. Lee (1972, p. 3). 2 1,1 Models In The Planning Process The use of computer models i n s c i e n t i f i c r e s e a r c h , business management, and government planning i s so widespread t h a t •modeling' has become a c o l l e c t i v e term f o r many d i f f e r e n t t e c hniques. For t h i s d i s c u s s i o n , a model of a d e c i s i o n problem w i l l be d e f i n e d as an a b s t r a c t i o n of the problem i n mathematical terms. Encoding of t h i s mathematical a b s t r a c t i o n i n a computer language c r e a t e s the computer model. 1 model prov i d e s a framework f o r the c o m p l e x i t i e s and u n c e r t a i n t i e s i n v o l v e d i n a d e c i s i o n making problem: - what i s the o b j e c t i v e or o b j e c t i v e s ? - what i s the range of a l t e r n a t i v e d e c i s i o n s t o be cons i d e r e d ? - what i s the i n f o r m a t i o n a v a i l a b l e t o e v a l u a t e the i m p l i c a t i o n s o f these d e c i s i o n s ? a b s t r a c t i o n i s the essence o f model b u i l d i n g . The modeller must capture the crux of the d e c i s i o n problem while l e a v i n g out the l e s s r e l e v a n t d e t a i l s . The process of a b s t r a c t i o n i s an i n t e g r a l p a r t of human problem s o l v i n g , and p e r c e p t u a l c a p a c i t y i s q u i c k l y exhausted by the b a s i c data of any but the s i m p l e s t d e c i s i o n problem: we de a l with stands, type groups or f o r e s t s , r a t h e r than t r e e s . As a r e s u l t of t h i s process of a b s t r a c t i o n , the model i s only an approximation to r e a l i t y , and any e r r o r s i n the c h o i c e of the elements of the model w i l l r e s u l t i n a biased approximation. Consequently, a s o l u t i o n to the mGdel d e c i s i o n problem r a r e l y c o n s t i t u t e s a s o l u t i o n t o the r e a l problem. I t i s 3 the r o l e of the manager to take i n s i g h t s gained i n the a n a l y s i s of the d e c i s i o n model and to use them to help s o l v e the r e a l management problem. The model i s a t o o l f o r a n a l y s i s , not d e c i s i o n making. C o n s i d e r a t i o n o f the planning process of the E r i t i s h Columbia F o r e s t S e r v i c e (BCFS) w i l l r e v e a l other d e s i r e a b l e c h a r a c t e r i s t i c s of a planning model. 1.2 The Planning Process Of The B r i t i s h Columbia F o r e s t S e r v i c e The Planning D i v i s i o n of the BCFS r e c o g n i z e s f o u r l e v e l s of planning (Figure 1) ; the cut block, the watershed, the management u n i t , and the r e g i o n . For completeness, the r o l e of the p r o v i n c i a l l e v e l of f o r e s t lands p l a n n i n g should be i n c l u d e d . The planning problem at each l e v e l i s c h a r a c t e r i z e d by one or more management o b j e c t i v e s and a s e t of management d e c i s i o n s , some of which are c o n s t r a i n e d by the neighbouring planning l e v e l s . For example, the o b j e c t i v e s of management u n i t planning are t o ensure t h a t f o r e s t resource production g o a l s f o r the u n i t , s e t at the r e g i o n a l l e v e l , are met. Land s t r a t i f i c a t i o n and use zoning t o g e t h e r with the l a y o u t and s c h e d u l i n g of the road system, are major d e c i s i o n s made at t h i s l e v e l . The land s t r a t i f i c a t i o n and use zoning both r e f l e c t r e g i o n a l land use g o a l s , and c o n s t r a i n the development of these g o a l s . For example, the p r o d u c t i v e c a p a c i t y of each land u n i t as measured Figure 1. Levels of f o r e s t planning, (from BCFS (1975) ). 5 by the annual a l l o w a b l e cut (AAC) c a l c u l a t i o n 2 i s necessary i n f o r m a t i o n f o r r e g i o n a l decisionmaking. The planning process d e s c r i b e d above can be c h a r a c t e r i z e d as a c h a i n of planning problems (Figure 2 ). At each planning l e v e l , seme of the d e c i s i o n s are l a r g e l y d e f i n e d or c o n s t r a i n e d by d e c i s i o n s made at other l e v e l s . These c o n s t r a i n i n g r e l a t i o n s h i p s are represented by arrows between the r e c t a n g u l a r d e c i s i o n spaces i n the accompanying f i g u r e . In a d d i t i o n t o d e c i s i o n l i n k s between the planning l e v e l s , c e r t a i n ' s t a t e ' f a c t o r s or i n f o r m a t i o n requirements transcend the plann i n g process. The p r o d u c t i v e c a p a c i t y of the f o r e s t lands must be con s i d e r e d at each l e v e l , although with varying d e t a i l . S i m i l a r l y , the f i s c a l s t a t e of the pr o v i n c e and the market p o t e n t i a l f o r f o r e s t products a f f e c t d e c i s i o n s at each l e v e l of the pl a n n i n g process. While many planning d e c i s i o n s at each l e v e l are h i g h l y c o n s t r a i n e d , seme d e c i s i o n s can be made f r e e l y . These 'degrees o f freedom' i n the d e c i s i o n problem allow use of s p e c i a l i z e d or l o c a l knowledge, and allow planners at each l e v e l the op p o r t u n i t y f o r optimum d e c i s i o n making, w i t h i n the bounds of the d e c i s i o n problem at th a t l e v e l . The m u l t i l e v e l s t r u c t u r e of the plann i n g problem a l s o a l l o w s planning d e c i s i o n s t o be time staged. P l a n n i n g d e c i s i o n s can be made at the lower ( o p e r a t i o n a l ) l e v e l s i n response to changing c o n d i t i o n s (e.g. short term market trends) as long as 2 For an e x p l a n a t i o n of the BCFS a l l o w a b l e cut c a l c u l a t i o n see Haley (1975). long range m a r k e t ft potentials, P rov ince Reg ion Managemen t U n i t D e v e l o p m e n t A r e a or T rea tment Unit Cu t B lock ph i losophy of use, etc. t7 b o u n d a r y c o n d i t i o n s e a c h r e c t a n g l e r e p r e s e n t s t h e s e t o f m a n a g e m e n t d e c i s i o n s f o r a p l a n n i n g l e v e l m o s t d e c i s i o n s a r e c o n s t r a i n e d b y o t h e r l e v e l s o f m a n a g e m e n t o n l y s o m e d e c i s i o n s a r e f r e e a t e a c h l e v e l 4. biophys ica l r e a l i t i e s , current m a r k e t s , etc. - b o u n d a r y c o n d i t i o n s F i g u r e 2 . A c o n c e p t u a l m o d e l o f t h e p l a n n i n g p r o c e s s . 7 they stay w i t h i n the d e c i s i o n and s t a t e c o n s t r a i n t s imposed , by the adjacent l e v e l s of p l a n n i n g . The n e c e s s i t y of a r a d i c a l d e c i s i o n that v i o l a t e s one of these c o n s t r a i n t s r e s u l t s i n an adjustment of planning d e c i s i o n s up and down the planning c h a i n . T h i s leads to another i m p l i c a t i o n of the nested m u l t i l e v e l d e c i s i o n problem model of the BCFS planning process. There i s no inst a n t a n e o u s or simultaneous s o l u t i o n to the o v e r a l l planning problem. One of the o b j e c t i v e s of p l a n n i n g must be to c r e a t e a d e c i s i o n making s t r u c t u r e that i s f l e x i b l e and r e s p o n s i v e to changing c o n d i t i o n s o f the d e c i s i o n environment such as markets, s o c i e t a l v a l u e s , or c a t a s t r o p h i c changes i n the f o r e s t l a n d base. The nested system of planning problems i s c o n s t a n t l y a d j u s t i n g d e c i s i o n s up and down the c h a i n i n response to changing c o n d i t i o n s . The purpose of the above h i g h l y s i m p l i f i e d c h a r a c t e r i z a t i o n of the planning process i s to show t h a t many of the q u a l i t i e s of the i d e a l process ( f l e x i b i l i t y , r e s p o n s i v e n e s s , dynamic nature) d e r i v e from s t r u c t u r e r a t h e r than the philosophy of f o r e s t l a n d management d e c i s i o n s 3 . As a . f i r s t step i n the 'process of a b s t r a c t i o n * , we see t h a t a planning model should r e f l e c t the s t r u c t u r e of the planning system as w e l l as c o n s i d e r i n g a l l the pl a n n i n g a l t e r n a t i v e s . T h i s i m p l i e s t h a t a model of a d e c i s i o n problem at any l e v e l of the system must have the c a p a b i l i t y to l i n k to the other l e v e l s of the planning process. The irodel 3 The author a p p r e c i a t e s that planning i n the BCFS may not occur i n p r a c t i c e as o u t l i n e d above. The d e s c r i p t i o n i s i n c l u d e d as an i d e a l i z e d example. 8 should not be 'separated' and optimized i n i s o l a t i o n a t each l e v e l . To optimize the s i n g l e l e v e l d e c i s i o n model i n the context of the the p l a n n i n g process, the d e c i s i o n and s t a t e l i n k a g e s can be c o n s i d e r e d as c o n s t r a i n t s . To choose d e c i s i o n s t h a t are o p t i m a l over m u l t i p l e planning l e v e l s , the s o l u t i o n technique must allow the e f f e c t s of the d e c i s i o n s to be t r a n s m i t t e d throughout the c h a i n of planning problems, and f o r each l e v e l to a d j u s t i t s d e c i s i o n s i n response. Most f o r e s t land planning models t r e a t a planning l e v e l as an i s o l a t e d subsystem of the planning process, or attempt to s o l v e m u l t i p l e l e v e l problems s i m u l t a n e o u s l y . a f t e r c o n s i d e r a t i o n of p e r t i n e n t planning models, t h i s t h e s i s w i l l present a f u r t h e r a n a l y s i s of the f o r e s t land planning problem. Planning models and a n a l y t i c a l techniques that e x p l i c i t l y r e c o g n i z e the m u l t i l e v e l nature of the p l a n n i n g problem w i l l be e x p l o r e d , S p e c i f i c a l l y , models of f o r e s t management u n i t s and stands w i l l be i n t e g r a t e d i n a framework amenable to d e c i s i o n a n a l y s i s . 9 2• 1 Review Of Fo r e s t Stand And Management Unit Models In Chapter 2, f o r e s t stand and management u n i t models w i l l be reviewed to i d e n t i f y the v a r i o u s l i n e s of development, and to examine the amount and nature of the i n f o r m a t i o n t h a t they provide about a planning problem. The models w i l l be grouped i n t o the c a t e g o r i e s of s i m u l a t i o n and o p t i m i z a t i o n . Simulation models g e n e r a l l y provide d e t a i l e d d e s c r i p t i v e i n f o r m a t i o n about a s i n g l e p o i n t i n p o l i c y space, i . e . the i m p l i c a t i o n s of a s i n g l e p o l i c y are p r e d i c t e d i n d e t a i l . Optimum seeking models provide r e l a t i v e i n f o r m a t i o n about the whole p o l i c y space - what i s the best p o l i c y i n terms of some c r i t e r i o n of performance. The d i s t i n c t i o n between s i m u l a t i o n and o p t i m i z a t i o n models i s b l u r r e d : an o p t i m i z a t i o n procedure may i n v o l v e repeated s i m u l a t i o n s . > 2.1 Stand Si m u l a t i o n Models Stand s i m u l a t i o n models are commonly subd i v i d e d i n t o three c a t e g o r i e s : s i n g l e t r e e d i s t a n c e dependent, s i n g l e t r e e d i s t a n c e independent, and whole stand models. Distance dependent models r e q u i r e the l o c a t i o n of the i n d i v i d u a l t r e e , while l o c a t i o n i s unnecessary f o r d i s t a n c e independent models. The seminal work of Newnham (1964) has r e s u l t e d i n an a c t i v e l i n e of development of s i n g l e t r e e d i s t a n c e dependent models, tfewnham's D o u g l a s - f i r (Pseudctsuga m e n z i e s i i (Mirb.) 10 France) model simulated the diameter growth of a stand of known s p a t i a l c h a r a c t e r i s t i c s . Crown co m p e t i t i o n of neighbouring t r e e s reduced the growth. The model was c a l i b r a t e d f o r normal d e n s i t y , unmanaged stands, l i m i t i n g the g e n e r a l i t y of the approach. A f t e r Newnham e s t a b l i s h e d the conceptual framework, others modified the model's components and improved the i n f o r m a t i o n output. l e e (1967) c a l i b r a t e d the model f o r Lodgepcle pine J P i n u s c o n t o r t a Douglas), added the f a c i l i t y to compute volumes and revenues, and attempted to put c o n f i d e n c e l i m i t s cu the r e s u l t s . B e l l a (197 0) modelled aspen (Populus tremuloides (Michx.)) of i r r e g u l a r s p a c i n g , and p r o j e c t e d height growth as w e l l as dbh. L i n (1969) developed a s p a t i a l c o m p e t i t i o n index and implemented i t i n a model of the dbh growth of Western hemlock ^Tsuga h e t e r o p h y l l a (Kaf.) Sarg.). A s u b s t a n t i a l m o d i f i c a t i o n of the s t r u c t u r e of the s i n g l e t r e e d i s t a n c e dependent model was demonstrated by M i t c h e l l (1969), M i t c h e l l ' s model of White spruce j P i c e a s l a u c a (Moench) Voss) i n v o l v e d height and dbh growth. Competition was modelled d i r e c t l y by growing branches of adjacent t r e e s u n t i l c o n t a c t . Ek and Monserud (197 4) have c r e a t e d a g e n e r a l i z e d s i n g l e t r e e , d i s t a n c e dependent system f o r mixed s p e c i e s and uneven-aged stands. The system was designed as a management t o o l , and has f a c i l i t i e s to simulate numerous s i l v i c u l t u r a l treatments, such as s i t e a l t e r a t i o n , t h i n n i n g and pruning. Many of the i d e a s and concepts d e s c r i b e d above were i n c o r p o r a t e d i n t o the system of Arney (1972) and brought to a high l e v e l of u t i l i t y by Arney (1974) and Hegyi (1974). Arney's model of D o u g l a s - f i r simulates r a d i a l whorl growth, e n a b l i n g the 11 s i m u l a t i o n of the e f f e c t s of c o m p e t i t i o n on bole shape. Hegyi adapted Arney's model f o r Jack pine J P i n u s banksiana Lamb,), i n c l u d i n g t h i n n i n g and f e r t i l i z a t i o n management a c t i o n s . Although i t would be r i s k y to d e s c r i b e Arney's model as the s t a t e of the a r t of s i n g l e t r e e d i s t a n c e dependent models, i t i s , at l e a s t , r e p r e s e n t a t i v e i n the nature and q u a l i t y of i n f o r m a t i o n i t produces. The s i m u l a t i o n of bole form a l l o w s d e t a i l e d i n f o r m a t i o n on l o g grades and g u a l i t y , e s t i m a t e s of s p e c i f i c g r a v i t y , r i n g s per i n c h , e t c , The easy c o n v e r s i o n of stand c h a r a c t e r i s t i c s to products f a c i l i t a t e s f i n a n c i a l e v a l u a t i o n of a stand. Accurate and d e t a i l e d i n f o r m a t i o n with which to e v a l u a t e a stand p o l i c y i s the major advantage of s i n g l e t r e e d i s t a n c e dependent models. The disadvantages of t h i s approach are the d i f f i c u l t i e s i n h e r e n t i n f i n d i n g b i o l o g i c a l l y m eaningful! i n d i c e s of c o m p e t i t i o n , the n e c e s s i t y of stem c h a r t s as i n p u t , and the high c c s t of computer p r o c e s s i n g of the s p a t i a l r e l a t i o n s h i p s w i t h i n stands. The l i n e of development of "single t r e e , d i s t a n c e independent stand models i s not as c l e a r as with the d i s t a n c e dependent models. P a i l l e (1970) simulated diameter growth of D o u g l a s - f i r independent of stand d e n s i t y , with the s i n g l e t r e e as the b a s i c u n i t . Goulding (1972) simulated D o u g l a s - f i r growth and produced a c c e p t a b l e r e s u l t s over a wide range of s i t e s , t h i n n i n g and spacing (Munrc, 1973). M o r t a l i t y was p a r t i a l l y s t o c h a s t i c - an awkward, i n e l e g a n t , but e f f e c t i v e s o l u t i o n to the problem of a l l o c a t i n g a continuous measure of m o r t a l i t y (basal area) among d i s c r e t e u n i t s (stems). Stage (1973) used a 12 s i m i l a r approach. C l u t t e r and A l l i s o n (1974) d e s c r i b e d a model t h a t i s not based on a s i n g l e t r e e jaer se, but i s the l o g i c a l e x t r a p o l a t i o n of the approach. Instead of mai n t a i n i n g a dbh l i s t , C l u t t e r and A l l i s o n approximated the diameter d i s t r i b u t i o n of the stems with the W e i b u l l d i s t r i b u t i o n . The p r o b a b i l i t y d e n s i t y f u n c t i o n was then d i v i d e d i n t o 25 c l a s s e s of egual p r o b a b i l i t y . Basal area increment and m o r t a l i t y i n terms of b a s a l area per acre and stems per acre are a l l o c a t e d among the v a r i o u s diameter c l a s s e s . T h i s f o r m u l a t i o n preserves much o f the d e t a i l of the s i n g l e t r e e approach but avoids m a i n t a i n i n g a leng t h y dbh l i s t . Eurkhart and Strub (1974) used a s i m i l a r approach t o model L o b l o l l y pine J P i n u s taeda L.) p l a n t a t i o n s , but approximated the diameter d i s t r i b u t i o n with a beta p r o b a b i l i t y d e n s i t y f u n c t i o n . S i n g l e t r e e d i s t a n c e independent models are much l e s s expensive to run than d i s t a n c e dependent models and dc not r e q u i r e stem maps as i n p u t , but g e n e r a l l y produce l e s s d e t a i l e d i n f o r m a t i o n . The f a c t t h a t C l u t t e r and A l l i s o n ' s model was developed as an o p e r a t i o n a l t o o l by New Zealand F o r e s t Products i s evidence of the u t i l i t y of the approach. The concept of whole stand, d i s t a n c e independent models encompasses a wide range of y i e l d p r e d i c t i o n t o o l s , i n c l u d i n g normal t a b l e s or e m p i r i c a l volume-age curves, as we l l as the r e g r e s s i o n function-based systems t o be c o n s i d e r e d here. Meyers (1971) has produced whole stand models that are used o p e r a t i o n a l l y by the U.S. Fores t S e r v i c e i n Wyoming and Colorado to manage pondercsa pine (Pinus jgonderosa Laws,). Meyers' model used r e g r e s s i o n f u n c t i o n s to p r e d i c t dbh, estimate h e i g h t . 13 estimate height and diameter i n c r e a s e from t h i n n i n g s , and p r e d i c t n o n - c a t a s t r o p h i c m o r t a l i t y . The model's a p p l i c a b i l i t y f o r managed stands i s enhanced by an a l g o r i t h m t o p r e d i c t s t o c k i n g a f t e r c u t t i n g . Stand d e n s i t y to be l e f t a f t e r each c u t t i n g i s expressed as a r e l a t i o n s h i p between b a s a l area and average stand diameter. R e s u l t s of t h i n n i n g s t u d i e s and data from temporary p l o t s are used to c o n s t r u c t a graph of d e s i r e d b a s a l area over average stand diameter f o r l o c a l average s i t e q u a l i t y . 'Best' stand d e n s i t y i s s e l e c t e d to r e f l e c t p r o d u c t i o n g o a l s . E n t e r i n g the curve with a diameter a f t e r t h i n n i n g r e t u r n s the a p p r o p r i a t e b a s a l area per acre. Meyers' (1973) model uses the same approach to stand growth but with the f a c i l i t i e s t o simulate timber management by shelterwocd, seed t r e e , or c l e a r c u t t i n g systems. The f i n a n c i a l y i e l d o f the stand i s si m u l a t e d , with r e p o r t s d e t a i l i n g p e r i o d i c c o s t s and r e t u r n s , and a n a l y s e s of r a t e earned. Hoyer (1975) used a whole stand d i s t a n c e independent model to e v a l u a t e D o u g l a s - f i r i n t e n s i v e management p r a c t i c e s . Increment f u n c t i o n s of gross b a s a l area and cubi c f o o t volume per year f o r thinned and unthinned stands form the b a s i s of the model, and r e s u l t s compare c l o s e l y t o Goulding's model. Hcyer's study i s an e x c e l l e n t example of the use of a r e l a t i v e l y simple stand s i m u l a t i o n model to analyse management d e c i s i o n s . Ek (1974) and Moser (1974) d e s c r i b e d systems of equations f o r s i m u l a t i n g growth i n diameter c l a s s e s . T h i s approach i s very s i m i l a r to the diameter d i s t r i b u t i o n f u n c t i o n model of C l u t t e r and A l l i s o n (1974), but i s i n c l u d e d here as a method f o r stand t a b l e p r o j e c t i o n . 14 Of the t h r e e c a t e g o r i e s of stand models reviewed, the whole stand p r o j e c t i o n approach r e q u i r e s the l e a s t data ( u s u a l l y l i t t l e more than c o n v e n t i o n a l i n v e n t o r y s t a t i s t i c s ) and i s the l e a s t expensive i n terms of computer p r o c e s s i n g . However, i t a l s o provides the l e a s t i n f o r m a t i o n f o r d e c i s i o n a n a l y s i s as i n d i v i d u a l t r e e data are t o t a l l y l a c k i n g . Management a c t i o n s and t h e i r responses are modelled at coarse l e v e l , and c o n v e r s i o n of y i e l d s to products f o r a p p r a i s a l i s d i f f i c u l t , A s o l u t i o n to the problem of t r a d i n g o f f d e t a i l f o r p r o c e s s i n g c o s t i s to use the a p p r o p r i a t e modelling approaches to simulate d i f f e r e n t time p e r i o d s of a management regime. For example, Adams and Ek (1976) suggested using a diameter d i s t r i b u t i o n f u n c t i o n model to p r o j e c t a stand from o r i g i n to a time (10-30 years of age) when s t r u c t u r e manipulation would be of i n t e r e s t . At t h a t time the diameter c l a s s models would be more a p p r o p r i a t e f o r p r o j e c t i o n s . Hegyi (1976) i s developing a compatible system of growth s i m u l a t i o n s which w i l l allow the user to s e l e c t the l e v e l of d e t a i l a p p r o p r i a t e at each time p e r i o d i n the simulated management seguence. Another approach i s to use d e t a i l e d models to c a l i b r a t e o ther models at a higher l e v e l of aggregation. Stage (1973) proposed use of a s i n g l e t r e e d i s t a n c e independent model t o c a l i b r a t e Meyers' whole stand p r o j e c t i o n system. 15 2.2 Stand O p t i m i z a t i o n Models Some attempts have been made to d e r i v e o p t i m a l management sequences through marginal a n a l y s i s (Chappelle and Nelson, 1964) but the most s u c c e s s f u l work to date has i n v o l v e d dynamic programming (DP) models. Amidon and Akin (1968) a p p l i e d dynamic programming to the problem of determining optimal l e v e l s of growing stock i n a stand d u r i n g a r o t a t i o n . The model's s t a t e i s the volume of saw timber per a c r e , and the stage i s stand age. R e s u l t s were comparable to Chappelle and Nelson, but the DP approach had computational advantages. As i s t y p i c a l with stand o p t i m i z a t i o n models, the component t h a t s i m u l a t e s the s t a t e t r a n s i t i o n , i . e . growth of saw timber, was extremely s i m p l i s t i c , c o n s i s t i n g of a volume increment eguation f o r thinned and unthinned stands. S i m i l a r l y , Schreuder (1971) used a volume increment growth f u n c t i o n i n a dynamic programming f o r m u l a t i o n to determine optimal t h i n n i n g and r o t a t i o n schedules f o r an even-aged f o r e s t . Naslund (1969) examined the same problem with the continuous maximum p r i n c i p l e , a technique c l o s e l y r e l a t e d to dynamic programming through m u l t i s t a g e a n a l y s i s . Naslund cn l y d e r i v e d the necessary c o n d i t i o n s f o r an o p t i m a l p o l i c y and d i d not present an e x p l i c i t s o l u t i o n technique. K i l k k i (1970) used a dynamic programming model to compute optimum c u t t i n g p o l i c i e s , again based on a volume increment f u n c t i o n . His model i s n o t a b l e , however, f o r i t s emphasis on s i m u l a t i n g the f i n a n c i a l aspects of stand management. The d e t e r m i n i s t i c o p t i m i z a t i o n models d e s c r i b e d above 16 c o n s t i t u t e a c l e a r l i n e of development. A number of r e s e a r c h e r s have attempted to simulate the s t o c h a s t i c nature of stand development i n an o p t i m i z a t i o n framework. A Markov chain model f o r p r e d i c t i n g diameter d i s t r i b u t i o n s was r e p o r t e d by Eudra (1S68), The model i s based on the assumed e x i s t e n c e of a s t a t i o n a r y (independent of age) matrix of p r o b a b i l i t i e s of t r a n s i t i o n from one diameter c l a s s t o another. The i m p l i c a t i o n of t h i s Markov assumption i s t h a t the p r o b a b i l i t y of a t r e e moving from the 4 i n c h dbh c l a s s to the 6 i n c h dbh c l a s s , i s independent of the age of the t r e e . A more r e a l i s t i c and management o r i e n t e d f o r m u l a t i o n i n v o l v i n g s t o c h a s t i c Markov processes was prepared by Hool (1966) and r e f i n e d by Lembersky and Johnson (1975). In Hcol's approach, the stand changes s t a t e a c c o r d i n g to f i x e d p r o b a b i l i t i e s . Lembersky and Johnson added the f a c i l i t y t o t r e a t the i n t e n s i t i e s of management a c t i o n s as d e c i s i o n v a r i a b l e s , with average dbh, stand d e n s i t y and market p r i c e as s t a t e v a r i a b l e s . The fundamental d i f f e r e n c e between the two models i s t h a t Hool r e l i e d cn f i n i t e d i s c r e t e dynamic programming methods, while Lembersky and Johnson more c o r r e c t l y t r e a t e d the time h o r i z o n as unbounded. Stand growth was simulated with equations p r e d i c t i n g average diameter and d e n s i t y at the end of a 10 year p e r i o d , as a f u n c t i o n of diameter and de n s i t y a t the beginning of the p e r i o d . The f i n a n c i a l y i e l d s were modelled with r e g r e s s i o n f u n c t i o n s of the number of t r e e s p l a n t e d , number of t r e e s t h i n n e d , and volume harvested. A f i n a l method to be reviewed was r e p o r t e d by Adams and Ek (1976). These authors are the f i r s t t o r e p o r t the o p t i m i z a t i o n 17 of a stand model c f g r e a t e r complexity than a s i n g l e whole stand p r o j e c t i o n . D e t e r m i n i s t i c diameter c l a s s models were i n c o r p o r a t e d i n t o a c l a s s i c a l mathematical programming f o r m u l a t i o n of the stand management problem. The r e s u l t i n g n o n l i n e a r system was optimized by a g r a d i e n t p r o j e c t i o n a l g o r i t h m . The authors r e p o r t e d c o n s i d e r a b l e d i f f i c u l t y i n converging to an optimal s o l u t i o n and suggested t h a t a n o n l i n e a r decomposition f o r m u l a t i o n ( G e o f f r i o n , 1970) should be explored. In summary, onl y the s i m p l e s t forms of f o r e s t stand s i m u l a t i o n have been s u c c e s s f u l l y i n c o r p o r a t e d i n t o o p t i m i z a t i o n models. A l l the f o r m u l a t i o n s reviewed s a c r i f i c e accuracy and d e t a i l c f p o l i c y s i m u l a t i o n f o r speed of e x e c u t i o n . 2.3 Management Unit S i m u l a t i o n Models The l i t e r a t u r e on management u n i t s i m u l a t i o n models i s not as e x t e n s i v e as t h a t of f o r e s t stand s i m u l a t i o n . Gould and 0'Began (1965) d e s c r i b e d the Harvard F o r e s t s i m u l a t o r . I t modelled growth, h a r v e s t i n g , s t o c h a s t i c c a t a s t r o p h i c events such as f i r e , and used c o s t and p r i c e data to prepare f i n a n c i a l r e p o r t s . C l u t t e r and Bamping (1965) d e s c r i b e d a s i m i l a r system, the F o r e s t Operations Simulator (FOPS), but i n v o l v i n g a l a r g e r l a n d base and more s o p h i s t i c a t e d b i o l o g i c a l and economic models. Formulas f o r p r e d i c t i o n of growth and y i e l d f o r the a p p r o p r i a t e s p e c i e s (planted and unplanted) were provided as i n p u t . The system was used to e v a l u a t e two management p o l i c i e s : an area 18 r e g u l a t i o n procedure and r e g u l a t i o n based on economic matu r i t y . The program produced e x t e n s i v e f i n a n c i a l r e p o r t s to a i d i n p o l i c y a n a l y s i s . The Purdue Management Game (Bare, 1970) was designed f o r e d u c a t i o n a l purposes but demonstrated the value of t h i s k i n d of s i m u l a t i o n f o r d e v e l o p i n g i n s i g h t i n t o management problems, Management u n i t s i m u l a t i o n models have been developed f o r o p e r a t i o n a l use by Meyers (1970, 1974). The Timber E v a l u a t i o n and Planning (TEVAP) system c a l c u l a t e s i n v e n t o r y summaries and p r e s c r i b e s management of even aged and two-staged f o r e s t stands. The system was e v a l u a t e d i n a 2-year o p e r a t i o n a l t e s t , r e p o r t e d by Edwards et a l . (1973). A p p l i c a t i o n of TEVAP r e s u l t e d i n s u b s t a n t i a l savings i n c o s t and time, due t o the i n c r e a s e d e f f i c i e n c y i n producing management plans. Other examples of o p e r a t i o n a l management u n i t s i m u l a t i o n systems are the S i m u l a t i n g I n t e n s i v e l y Managed Allowable Cut (SIMAC) system of Sassman et a l . (1972) and the Allowable Cut P r o j e c t i o n (ACP) system of the B.C. F o r e s t S e r v i c e (McPhalen, 1976) . The systems d e s c r i b e d above were designed f o r timberlands management. Recently, s i m u l a t i o n systems have been c o n s t r u c t e d t o a i d i n planning the i n t e g r a t e d use of f o r e s t l a n d s . The Snohomish V a l l e y Environmental Network (SVEN; Bare and Schreuder, 1976) uses a set of s i m u l a t i o n models l i n k e d to a geographic data base to examine the p h y s i c a l , economic, and environmental consequences of a l t e r n a t i v e w i l d l a n d use d e c i s i o n s . The system has s u b s t a n t i a l data needs: i n v e n t o r y data t o i n i t i a l i z e the resource data base, and e m p i r i c a l response 19 data necessary t o c a l i b r a t e the subsystem models. The M i d l a n d s Resource A l l o c a t i o n Procedure (WRAP; Ha inner, 1976) i s a system developed by the Tennessee V a l l e y A u t h o r i t y to a s s i s t p r i v a t e landowners to manage t h e i r woodlands. The system c o n s t r u c t s a timber h a r v e s t i n g schedule which w i l l produce a number cf m u l t i p l e use b e n e f i t s . Management u n i t s i m u l a t o r s r a r e l y model stand growth by any but the s i m p l e s t methods, However, the f i n a n c i a l components of the problem are o f t e n handled i n d e t a i l and the r e s u l t i n g i n f o r m a t i o n i n r e p o r t s and summaries can be voluminous. Because of the crude l e v e l of stand s i m u l a t i o n , these systems are g e n e r a l l y inexpensive to run. Conseguently, d e c i s i o n a n a l y s i s with t h i s type of model r e g u i r e s choosing s e t s of p o l i c i e s to simulate and examining the d e t a i l e d output to decide on the •best' p o l i c y . I f the d e f i n i t i o n of what c o n s t i t u t e s a 'best' p o l i c y i s e a s i l y expressed i n f u n c t i o n a l form, then a s e a r c h i n g cr o p t i m i z i n g procedure i s a t t r a c t i v e . 20 2.4 Management Un i t O p t i m i z a t i o n Models O p t i m i z a t i o n models f o r management u n i t planning have been r a p i d l y developed and implemented o p e r a t i o n a l l y f o r many p u b l i c and p r i v a t e f o r e s t e n t e r p r i s e s . C u r t i s (1962) proposed a l i n e a r programming (LP) model f o r the management of f o r e s t property. Kidd e t a l . (1966) d e s c r i b e d a case study of f o r e s t r e g u l a t i o n by l i n e a r programming. These e a r l y attempts at a p p l y i n g LP t o f o r e s t management problems were u s u a l l y l i m i t e d by the l a r g e s i z e of the l i n e a r model, r a t h e r than a c t u a l computer p r o c e s s i n g time. L i n e a r programming decomposition f a c i l i t i e s were a p p l i e d to the management u n i t problem by L i i t t s c h w a g e r and Tcheng (1967), i n an attempt to overcome these s i z e d i f f i c u l t i e s . N a u t i y a l and Pearse (1966) demonstrated how l i n e a r programming techniques can be used to s p e c i f y the economically optimum p a t t e r n of ha r v e s t s from an i r r e g u l a r f o r e s t during the p e r i o d of i t s conver s i o n to s u s t a i n e d y i e l d . The 'normal* f o r e s t i s the t a r g e t s t a t e . The authors concentrated on u s i n g the l i n e a r model to analyse economic r e l a t i o n s h i p s i n the management u n i t planning problem. N a u t i y a l (1966) used the same l i n e a r model to compute •user c o s t ' , a measure of temporal e f f i c i e n c y i n timber h a r v e s t i n g , e s s e n t i a l y o p p o r t u n i t y c o s t i n time dimension. In another paper recommending the a n a l y t i c c a p a b i l i t i e s of l i n e a r programming, Navon and McConnen (1967) d e s c r i b e d the use of the post o p t i m a l technique of parametric programming f o r e v a l u a t i n g f o r e s t management p o l i c i e s . T h i s technique a l l o w s the 21 s y s t e m a t i c e x p l o r a t i o n of the p o l i c y space i n the neighborhood of the optimum p o l i c y . One of the f i r s t IP models to be used e x t e n s i v e l y f o r managing i n d u s t r i a l f o r e s t s was developed by Ware and C l u t t e r (1971), The MAX-MILLION ( C l u t t e r , 1968) system was developed from the e a r l i e r FOPS s i m u l a t o r ( C l u t t e r and Baraping, 1965) and generates an LP matrix, A commercial LP s o l u t i o n system ' s o l v e s ' the matrix and s e l e c t s the optimum se t of management a l t e r n a t i v e s . A t h i r d program, PROPHET (Fortson, 1970), performs a cash flow a n a l y s i s cn the optimum p o l i c y produced by MAX-KILLICN and the LP system. Navon (1971) c r e a t e d a s i m i l a r system f o r p u b l i c t i m b e r l a n d s , the Timber Resource A l l o c a t i o n Method (RAM). The RAM matrix generator uses volume and f i n a n c i a l y i e l d t a b l e s to generate a l t e r n a t i v e sequences of management a c t i o n s f o r timber c l a s s e s . The management a l t e r n a t i v e s and l i n e a r model are processed by a commercial IP system. A r e p o r t w r i t e r d i s p l a y s the o p t i m a l p o l i c y i n t a b l e s , summaries and graphs. Timber RAM i s used e x t e n s i v e l y by the U.S. F o r e s t S e r v i c e and has been proposed f o r use by the B.C. F o r e s t S e r v i c e (Williams e t a l . , 1975). An e x t e n s i o n of the timber management model to i n c l u d e t r a n s p o r t a t i o n a c t i v i t i e s has been r e c e n t l y d e s c r i b e d by Weintraub and Navon (1976). L i n e a r programming models have proven most a t t r a c t i v e f o r d e c i s i o n a n a l y s i s at the management u n i t l e v e l , although the l i n e a r model has recognized inadequacies. The most s e r i o u s f a u l t o f the l i n e a r model as a p p l i e d to management u n i t planning i s the n e c e s s i t y to c o n s t r a i n volume flow to o b t a i n r e a l i s t i c 22 s o l u t i o n s , fin o p t i m i z a t i o n model l i m i t e d by a downward s l o p i n g demand curve would be more r e a l i s t i c i n many s i t u a t i o n s . However, the demand curve approach d e s t r o y s the l i n e a r s t r u c t u r e of the model 4. i a l k e r (1976) presented an a l t e r n a t i v e t o l i n e a r models, the Economic Harvest O p t i m i z a t i o n Model (ECHO) t h a t i s e s s e n t i a l l y an e x t e n s i o n of the Faustman c r i t e r i a of f i n a n c i a l m a t u r i t y . Negatively s l o p e d demand curves and/or p o s i t i v e l y s l o p e d marginal c o s t curves are used to c a l c u l a t e timber h a r v e s t s . A binary search technique i s used t o i d e n t i f y the p e r i o d i c harvest volume where the o p t i m i z a t i o n c r i t e r i a h o lds. HcDonough and Park (1975) d e s c r i b e d Walker's s o l u t i o n technigue as 'r a t h e r ad hoc', and ref o r m u l a t e d the problem as an op t i m a l c o n t r o l problem u t i l i z i n g the d i s c r e t e maximum p r i n c i p l e . The model i s too simple f o r o p e r a t i o n a l use but i s an e x c e l l e n t demonstration of the a p p l i c a t i o n of the mathematics of c o n t r o l systems to f o r e s t management models. Another c r i t i c i s m of the l i n e a r model i s t h a t only a small p r o p o r t i o n of a l l the p o s s i b l e a l t e r n a t i v e management sequences can be co n s i d e r e d (Johnson and Scheurman, 1974). Nazareth (1973) d e s c r i b e d a l i n e a r programming decomposition model that i m p l i c i t l y c o n s i d e r s a l l f e a s i b l e management seguences but maintains only a s m a l l s e l e c t i o n i n the LP matrix. The work of Nazareth was brought to the author's a t t e n t i o n during the course o f t h i s study, and i s very s i m i l a r to elements of the model * The demand curve can be i n c o r p o r a t e d i n t o the l i n e a r model d i r e c t l y with separable programming, but with s u s b t a n t i a l computational burden. 23 d e s c r i b e d i n Chapter 5, which were developed independently. 2.5 Summary For e s t lands p l a n n i n g models at the stand and management u n i t l e v e l s are w e l l developed and widely o p e r a t i o n a l . Stand s i m u l a t i o n models of v a r i o u s degrees of aggregation can provide d e t a i l e d d e s c r i p t i v e i n f o r m a t i o n about a stand management p o l i c y . Stand o p t i m i z a t i o n technigues g e n e r a l l y employ only the s i m p l e s t stand s i m u l a t i o n , s a c r i f i c i n g d e s c r i p t i v e i n f o r m a t i o n f o r ease of computation. Management u n i t s i m u l a t i o n models have been combined with o p t i m i z a t i o n systems ( u s u a l l y i n a l i n e a r model) to c r e a t e a powerful t o o l f o r a n a l y s i s o f management u n i t d e c i s i o n problems. However, only the s i m p l e s t stand s i m u l a t i o n s are used to generate y i e l d i n f o r m a t i o n . The f i e l d i s r i p e f o r s y n t h e s i s and i n t e g r a t i o n . 24 3» Problem A n a l y s i s - The Process Of A b s t r a c t i o n 3.1 Terminology And Notation Various terms are proposed throughout the l i t e r a t u r e to d e s c r i b e the elements of mathematical programming models of the f o r e s t lands planning problem. In g e n e r a l , the use c f management terms w i l l f e l l o w t h a t of the Timber BAM system (Navon, 1971) as modified by H i l l i a m s et a l A f o r the BCFS Computer A s s i s t e d Resource Planning (CABP) p r o j e c t . The mathematical programming n o t a t i o n w i l l g e n e r a l l y r e l y on the c o n s t r a i n e d d e r i v a t i v e approach c f H i l d e and B e i g h t l e r (1967). Terms and n o t a t i o n w i l l be i n t r o d u c e d as needed throughout the paper. 5 At t h i s p o i n t some b a s i c terms must be d e f i n e d and n o t a t i o n assigned to prepare the way f o r c a s t i n g the f o r e s t lands planning problem i n mathematical programming format. s Symbols and n o t a t i o n are summarized i n Appendix I. 25 3.1,1 Land U n i t s The f o r e s t land area w i l l be co n s i d e r e d a t th r e e l e v e l s : the management u n i t , the treatment u n i t , and the type i s l a n d . The m an a g e me n t una, t i s simply the t o t a l f o r e s t area t o be sub j e c t e d to the planning a n a l y s i s . In the context of management i n B r i t i s h Columba, a management u n i t would be a P u b l i c S u stained Y i e l d U n i t (PSYU) or Tree Farm L i c e n c e (TFL). The tjjge i s l a n d s are l a n d u n i t s t h at are homogenous with r e s p e c t to some d e s c r i p t i v e v a r i a b l e s , such as f o r e s t cover, s o i l - l a n d f o r m c l a s s i f i c a t i o n , and designated use. The type i s l a n d i s contiguous and i s u s u a l l y formed by o v e r l a y mapping. A treatment u n i t i s an aggregation of type i s l a n d s on the management area having s i m i l a r management c h a r a c t e r i s t i c s . Hence, treatment u n i t s should be su s c e p t a b l e to the same range of management a c t i o n s . In terms of the stand s i m u l a t i o n model, the treatment u n i t i s the area of l a n d on which the management of the timber resource can be simulated with one s e t t i n g of the model parameters. Let i be a type i s l a n d i n management u n i t E, and u a treatment u n i t or aggregation of type i s l a n d s i n E, Then i' C'-E , i u C E The p a r t i t i o n i n g of E i n t o treatment u n i t s t h a t i s optimal f o r a s p e c i f i c land management model, can be approximated by the methods of Williams and Yamada (1976). Computer a s s i s t a n c e i s d e s i r a b l e t o form the treatment u n i t s , A t y p i c a l example d e s c r i b e d by Wi l l i a m s and Yamada i n v o l v e d the p a r t i t i o n i n g of 26 2441 type i s l a n d s i n t o 87 treatment u n i t s . 3.1.2 Management A c t i v i t i e s Management a c t i v i t i e s are assumed to occur i n d i s c r e t e i n t e r v a l s of time. Let the plann i n g horizon be p a r t i t i o n e d i n t o T d i s c r e t e i n t e r v a l s (t = 1,2,..., T) where the i n t e r v a l s are not n e c e s s a r i l y of egual l e n g t h . T h i s p a r t i t i o n i n g o f the h o r i z o n i n t o a s e t of d i s c r e t e i n t e r v a l s c r e a t e s the time frame of the system. The management a c t i v i t y on a treatment u n i t i s d e s c r i b e d at two l e v e l s , management a c t i o n s and management sequences. A management a c t i o n i s any a c t i o n t h a t can be a p p l i e d over a treatment u n i t . For example, p l a n t i n g , spacing and c l e a r c u t t i n g are management a c t i o n s . Let be a management a c t i o n at time t cn treatment u n i t u. Then the s e t of a l l a c t i v i t i e s f e a s i b l e on u at t i s A . . u t A management sequence - i s a seguence of management a c t i o n s extending over the whole of the planning h o r i z o n . Let a u be a management sequence and A^  be the s e t of a l l f e a s i b l e management sequences cn treatment u n i t u. The set Ain may be r e f e r r e d to as the set of a l t e r n a t i v e u f o r u. The k t h a l t e r n a t i v e management sequence i s a (k) u 27 3.1.3 State And State T r a n s i t i o n The management s t a t e of a treatment u n i t i s a set of d e s c r i p t i v e v a r i a b l e s which provide a l l the i n f o r m a t i o n necessary to make immediate and f u t u r e management d e c i s i o n s . A commodity s t a t e r e p r e s e n t s the s t a t u s c f some c c n s t r a i n e d commodity which must be a l l o c a t e d across the management u n i t . For example, a t y p i c a l l y c c n s t r a i n e d commodity i n f o r e s t lands p l a n n i n g problem i s the volume of timber cut i n a p a r t i c u l a r time perio d ( i . e . y i e l d c o n t r o l c o n s t r a i n t s ) . A f o r e s t stand s i m u l a t i o n model i s a mathematical approximation of how the s t a t e v a r i a b l e s change i n response to management a c t i o n s , i n a given time i n t e r v a l . At any i n s t a n t of simulated time, the model i s i n a p a r t i c u l a r s t a t e as d e f i n e d by the s t a t e v a r i a b l e s . I f the model i s a v a l i d a b s t r a c t i o n of the r e a l system, then t h i s s i m u l a t i o n s t a t e approximates the management s t a t e of the treatment u n i t . For example, i n Goulding's d i s t a n c e independent stand s i m u l a t i o n model, the stand age and a l i s t of stand diameters at breast height comprise the s t a t e . The i n f e r e n c e i s t h a t the age and diameter d i s t r i b u t i o n of a treatment u n i t provide s u f f i c i e n t i n f o r m a t i o n f o r d e c i s i o n making. The s i m u l a t i o n model c o n s i s t s of a f u n c t i o n a l r e l a t i o n s h i p between s t a t e and d e c i s i o n v a r i a b l e s , and model parameters. D e c i s i o n v a r i a b l e s can be manipulated f r e e l y while the s t a t e v a r i a b l e s a d j u s t i n response to changes in the d e c i s i o n s . For the f o r e s t land s i m u l a t i o n model, the d e c i s i o n v a r i a b l e s are the management a c t i o n s . The model parameters can be thought of as 28 cons t a n t s t a t e v a r i a b l e s t h a t have f i x e d values throughout the s i m u l a t i o n (e.g. s i t e , s p e c i e s type, s o i l c l a s s i f i c a t i o n ) . As a s t a t e t r a n s i t i o n - f u n c t i o n the s i m u l a t i o n model p r e d i c t s the f u t u r e s t a t e s as a f u n c t i o n o f the c u r r e n t s t a t e and management a c t i o n s . I f the model, c a l i b r a t e d with parameters a p p r o p r i a t e to treatment u n i t u i s M^, then the p r e d i c t e d s t a t e a t t+1 due to management a c t i o n a i s {v _ L F C } = M ({v .c ),a ) (3.1-1) u,t+l u,t+l u ut ut ut where v i s the management s t a t e and c . i s the amount of ut a ut commodity produced or consumed. The commodity i n p u t - o u t p u t i s a vector of J elements ( c l u t > c 2 u t , .. ., c j u t ) r e p r e s e n t i n g the J commodities of i n t e r e s t . The management s t a t e v a r i a b l e may s i m i l a r l y be m u l t i - d i m e n s i o n a l . The c o l l e c t i o n of the v a r i a b l e s i n t o higher l e v e l a r r a y s w i l l be s i g n i f i e d by the omission of the a p p r o p r i a t e s u b s c r i p t . For example, c i s the JxOxT matrix of commodity values c\ u f c . 29 3.1.4 Management O b j e c t i v e s And Beturns In a mathematical programming problem the o b j e c t i v e must be p r e c i s e l y s t a t e d . T y p i c a l management o b j e c t i v e s f o r a management u n i t are to maximize volume, maximize present net worth, or minimize c o s t s . The economic aspects of v a r i o u s o b j e c t i v e s used i n f o r e s t management were reviewed by Be n t l y and Teegarden (1965). A c o s t or p r o f i t i s a s s o c i a t e d with each management a c t i o n on a treatment u n i t . This a c t i o n r e t u r n B t i s a f u n c t i o n of both the management a c t i o n and the management s t a t e when the a c t i o n was taken. The r e t u r n f u n c t i o n used throughout t h i s work w i l l be the present net value f u n c t i o n R = R ( v .c .a ) = P ( v ,a .) / ( 1 + e ) Y ( t ' t 0 ) (3.1-2) ut ut ut ut ut ut ut ut where P i s the net p r o f i t c f a p p l y i n g a c t i o n a ^ to treatment u n i t u while i n s t a t e f v .,c^ ] , 8 i s the di s c o u n t r a t e , and L ut ut J y ( t , t ) i s number of years to be discounted over. A management sequence r e t u r n B i s formed by summing a l l the a c t i o n r e t u r n s encountered i n a management seguence. R = R ( v , c ,a ) = I R (3.1-3) u u u u u t = ^ u t ; The o b j e c t i v e f u n c t i o n f o r the whole management u n i t U i s simply the sum of a l l the management seguence r e t u r n s . 30 R = R ( v , c , a ) = E R u = l v (3.1-4) 3,1.5 Management C o n s t r a i n t s Management c o n s t r a i n t s can be grouped i n t o three c a t e g o r i e s : r e s t r i c t i o n s cn the a l l o c a t i o n (production or consumption) of some commodities of i n t e r e s t a c r o s s the management u n i t , s t r u c t u r a l r e l a t i o n s h i p s between the management s t a t e s of the treatment u n i t over time, - simple r e s t r i c t i o n s on the choice o r i n t e n s i t y of a management a c t i o n . The f i r s t category of c o n s t r a i n t s can be thought of as c o n s t r a i n t s 'bet ween 1 treatment u n i t s , while the l a s t two c a t e g o r i e s are c o n s t r a i n t s ' w i t h i n ' treatment u n i t s . For the f o r e s t planning problem, the commodity a l l o c a t i o n c o n s t r a i n t s are u s u a l l y d e r i v e d from c o n s i d e r a t i o n s such as y i e l d r e g u l a t i o n p o l i c i e s , cash flow reguirements, and nursery requirements f o r p l a n t i n g stock. For example, a t y p i c a l c o n s t r a i n e d commodity i n Timber RAM i s the volume flow requirement that the cut i n some p a r t i c u l a r time p e r i o d must not exceed seme upper cr lower bound. These commodity a l l o c a t i o n 31 c o n s t r a i n t s w i l l be rep r e s e n t e d i n gen e r a l form as e q u a l i t i e s ; g. (c. l = 0 (3.1-5) D Dut f o r a l l J c o n s t r a i n t commodities. The second category of c o n s t r a i n t s r e s t r i c t s the f l u c t u a t i o n of the management s t a t e s i n response t o a management a c t i o n . These r e l a t i o n s h i p s are u s u a l l y embodied i n the treatment u n i t s i m u l a t i o n model, but w i l l be represented i n equation form as i, / N (3.1-6) h (v ,a ) = 0 qu u u f o r a l l U treatment u n i t s and Q management s t a t e dimensions. The t h i r d category of c o n s t r a i n t s determines the elements of a , the set of a l l management a c t i o n s f e a s i b l e on treatment u n i t u at time t . C o n s t r u c t i o n of a must take i n t o account r e s t r i c t i o n s on the amount or i n t e n s i t y o f management a c t i o n s . For example, c o n s i d e r a r e s t r i c t i o n on the i n t e n s i t y of t h i n n i n g with the d e c i s i o n v a r i a b l e a d e f i n e d as the r a t i o of stand average diameter a f t e r t h i n n i n g , to average diameter before t h i n n i n g . B.C. Fo r e s t S e r v i c e g u i d e l i n e s 6 r e s t r i c t a t to the i n t e r v a l (.8,.95). In g e n e r a l , these d e c i s i o n r e s t r i c t i o n s w i l l be represented by & P r o v i s i o n a l allowances f o r i n t e n s i f i e d stand management p r a c t i c e s f o r use i n al l o w a b l e cut c a l c u l a t i o n s , B.C.F.S., 1972. 32 a £ A ut ^ ut (3.1-7) f o r a l l 0 treatment u n i t s and T time i n t e r v a l s . I f the d e c i s i o n and s t a t e v a r i a b l e s meet the r e s t r i c t i o n s Eqs, (3.1.5) - (3,1-7), then the management a c t i o n s comprise a f e a s i b l e s o l u t i o n , a f . 3.2 General Problem Formulation The timberlands p l a n n i n g problem can now be expressed i n terms of the r e l a t i o n developed i n s e c t i o n 3.1. Find a** = { a u | u<£E} such that * * * R = R(c,v,a ) (3.2-1) i s an optimum and that g. (c. ) = o D Dut j — 1 , . . . , J (3.2-2) h (v ,a ) = 0 qu u u q = 1 ,... ,Q u = 1 ,.. . ,U (3.2-3) a A ut ut (3.2-4) One approach to s o l v i n g the above problem i s to separate 33 the g e n e r a l problem i n t o two e a s i e r problems; an ' o u t e r 1 problem of a l l o c a t i n g the commodities of i n t e r e s t through s e l e c t i n g management seguences frcm a subset of p o s s i b l e management seguences, and an 'inner* problem of c r e a t i n g the a l t e r n a t i v e management sequences. The outer and inner problems w i l l be r e f e r r e d to as MP1 and MP2 r e s p e c t i v e l y . To formulate MP 1, l e t a be a candidate set of management sequences, c o n s t r u c t e d such t h a t i t c o n t a i n s a t l e a s t one f e a s i b l e s o l u t i o n £: . I t i s expected t h a t a w i l l g e n e r a l l y be much sm a l l e r than the complete s e t of a l l p o s s i b l e management sequences, A. (If any management a c t i o n can take on continuous v a l u e s , A w i l l not be f i n i t e ) . Then the commodity subprcblem can be expressed as f o l l o w s : 13P1: Find a* C a such t h a t z = R(c,a) (3.2-5) i s a maximum, and t h a t g.(c.) =0 j = 1,...,J (3.2-6) MP1 i s much e a s i e r to s o l v e than the g e n e r a l problem as the management seguences t o be considered are pr e - d e f i n e d and l i m i t e d to the r e s t r a i n e d set a. Of course, * ** Z(a ) < Z(a ) (3.2-7) 34 and another s e r i e s of MP2 problems must be solved t o provide improved management seguences to a . * The problem of f i n d i n g an optimal management seguence a i s not d i f f i c u l t when the treatment u n i t s are con s i d e r e d s e p a r a t e l y . MP2: For each treatment u n i t u, f i n d the sequence of management a c t i o n s a = ( a , , a „ , . . . , a m ) such t h a t ; ^ u u l ' u 2 ' u T y = R ( v , c , a ) (3.2-8) U XL U U i s maximized and t h a t * h ( v , a ) = 0 u q u u u = 1 , . . . , U (3.2-10) t = 1 , . . . , T U = 1 , . . . , U q = 1 , . . . , Q (3.2-9) * A 4 - A -u t u t However, because the commodity a l l o c a t i o n c o n s t r a i n t s of MP1 are not c o n s i d e r e d i n the MP2 problem, a s e t of management * * * sequences a 1 , a 2 , t h a t are optimal f o r MP2, may not be f e a s i b l e i n MP1. The r e l a t i o n s h i p between MP1 and MP2, and the proper f o r m u l a t i o n of MP2 as a subproblem of MP1, w i l l be examined i n depth l a t e r i n t h i s chapter. D i r e c t o p t i m i z a t i o n of the g e n e r a l problem i s u s u a l l y i m p o s s i b l e except a f t e r s i m p l i f y i n g assumptions or f o r h y p o t h e t i c a l management s i t u a t i o n s . T h i s t h e s i s w i l l examine and 35 develop s o l u t i o n s t r a t e g i e s based on a decomposition approach: the two l e v e l s o l u t i o n of a commodity a l l o c a t i o n problem and a s e t of o p t i m a l management sequence subprcblems. In the next s e c t i o n , the f o r e s t l ands planning problem i s examined with m u l t i s t a g e a n a l y s i s to d i s p l a y i t s u n d e r l y i n g mathematical s t r u c t u r e and to demonstrate i t s p o s s i b i l i t i e s f o r mathematical decomposition. 3.3 M u l t i s t a g e A n a l y s i s I t w i l l be shown to be advantageous to formulate the f o r e s t management problem as a s e r i a l m u l t i s t a g e system. A s e r i a l m u l t i s t a g e system c o n s i s t s of a s e r i e s of d e c i s i o n s t h a t i s arranged so t h a t each d e c i s i o n a f f e c t s the circumstances under which the next one i n the sequence must be made. In these s e q u e n t i a l problems, the input to any stage i s the output of the previous stage, and thus a f f e c t s a l l those f o l l o w i n g . The o b j e c t i v e of t h i s type of f o r m u l a t i o n i s t o optimize l a r g e systems p a r t i a l l y , one stage at a time. 36 3.3.1 ME1 As A 2-point Boundary Value Problem The MP1 d e c i s i o n problem i s to s e l e c t a set of management sequences * a 2 ' •••» au) such t h a t c e r t a i n c o n s t r a i n e d commodities are a l l o c a t e d across the management u n i t i n optimal manner. The formulaion of MP1 as a s e r i a l system r e q u i r e s the e x p l i c i t i n c o r p o r a t i o n of the commodity c o n s t r a i n t s . The a l l o c a t i o n must s t a r t with known commodity l e v e l s and must end with the commodity l e v e l s i n f e a s i b l e r e g i o n . Stage: u, u=1,2, .«., U At stage u of the s e r i a l system, a management sequence i s s e l e c t e d f o r treatment u n i t u D e c i s i o n v a r i a b l e : a C A u u The d e c i s i o n v a r i a b l e i s the management seguence a s e l e c t e d from the candidate set a . u State v a r i a b l e s : C = (C. , C. , c ) u l u 2 u J u (3.3-1) The i n p u t s t a t e C at stage u i s the J dimensional v e c t o r u of t o t a l commodity l e v e l s p r i o r to s c h e d u l i n g u. U - l T C . = I E c . j = l , . . . , j (3.3-2) : u i = 1 t D u t The i n i t i a l value of the t o t a l commodity s t a t e i s known C., and i s u s u a l l y zero f o r a commodity t h a t i s to be 37 produced on each treatment u n i t (e.g. volume). The boundary value has a p o s i t i v e value when a commodity i s consumed (e.g. f u n d i n g ) . c . . = c ° j = 1 J <3'3-3> The f i n a l value of the t o t a l commodity s t a t e i s u s u a l l y c o n s t r a i n e d between upper and lower bounds. < ~ < (3.3-4) L B . 1 C. _ UB. \ * I S t a t e t r a n s i t i o n f u n c t i o n : T C. = W. (C. ,a ) = C. + i c . j = 1 , . . . , J (3.3-5) ] u : u ] u u ju t = 1 ] U t The t r a n s i t i o n f u n c t i o n at stage u computes the t o t a l commodity s t a t e r e s u l t i n g from the a p p l i c a t i o n of management sequence a to treatment u n i t u. The i n c i d e n c e i d e n t i t y i s C = c. _t. (3.3-6) ] U ] , u + l Heturn f u n c t i o n : R = R ( C U ' C U - 1 C l ' a i ' a 2 V = E R (C ,a ) ( 3 ' 3 " 7 ) n u u u u = l The stage r e t u r n from managing u with a i s the u management seguence r e t u r n s d e s c r i b e d i n s e c t i o n 3.1.4. The t o t a l r e t u r n or o b j e c t i v e f u n c t i o n d e f i n e d over the 38 whcle management u n i t U i s the sum of a l l the management seguence r e t u r n s s e l e c t e d as elements of a. As the i n i t i a l t o t a l commodity s t a t e and the f i n a l t o t a l commodity s t a t e are f i x e d , MP2 i s a two-point boundary value problem. Boundary value problems always r e g u i r e d e c i s i o n i n v e r s i o n , the mathematical exchange of the r o l e s of d e c i s i o n v a r i a b l e and the output s t a t e v a r i a b l e i n the t r a n s i t i o n f u n c t i o n . T h i s i m p l i e s that c. = w. (c. ,a ) (3.3-8) j u j u j u u can be so l v e d f o r a u i n terms of C u and C u , to give a = w! (c ,c ) (3.3-9) u 311 u u Eg. (3.3-9) shows what d e c i s i o n i s needed to transform C to C , u u The computational f e a s i b i l i t y of t h i s approach f o r MP1 can be ig n o r e d f o r now but w i l l be examined i n s e c t i o n 3.4.1. The management u n i t o b j e c t i v e f u n c t i o n can be made to depend only cn the i n i t i a l s t a t e C , the f i n a l s t a t e C , and l u the U-1 remaining d e c i s i o n s , by u s i n g the s t a t e , t r a n s i t i o n f u n c t i o n and i t s d e c i s i o n i n v e r s e form. F i r s t , the f i n a l 0 d e c i s i o n i s r e p l a c e d with the U output s t a t e i n the o b j e c t i v e f u n c t i o n (Eg. (3.3-7)). R " R ( c u ' c u - l " " " ' c i ' a i : ' a 2 " " , a u - i f C u ) ( 3 * 3 1 0 ) Using the t r a n s i t i o n f u n c t i o n W , the s t a t e s C ,C , ,..,C can u 2 3 u 39 be e l i m i n a t e d , R = R ( w u _ i ( - " w 2 ( w i ( c i ' a i ) ' a 2 ) ' • • • ' a u _ l ) ' V (3.3-11) and the d e c i s i o n i n v e r s e form of the t r a n s i t i o n f u n c t i o n W provides the f i n a l d e c i s i o n a^ R = R ( W u . ( W u _ 1 ( . . . W 2 ( W 1 ( C 1 , a 1 ) , a 2 ) , . . . , a u _ 1 ) , C ) = R ( C 1 , a 1 , a 2 , . . . , a u _ 1 , ' G u ) (3.3-12) Thus, the two-point boundary value o p t i m i z a t i o n problem i s to f i n d the management sequences (a 1 # a 2 , a^ ) such t h a t the boundary c o n d i t i o n s are met, ] i 3 L B . < c. < U B . j = l , . . . , j (3.3-13) 3 - DU - j the s t a t e t r a n s i t i o n f u n c t i o n s , expressed as e q u a l i t y c o n s t r a i n t s , are met, c. -w (c. , a ) = o j = I , . . . , J (3.3-14) 3 ' U + 1 J U 3u u J = 1 D the manaqement sequences are f e a s i b l e , a c A u = l , . . . , u (3.3-15) u ^ — u and the management u n i t r e t u r n f u n c t i o n i s maximized: R = Maximum R(C ,a ,a 2» ... ' ^ ^ ' ^ a 1,...,a u (3. 3-16) The s e q u e n t i a l s t r u c t u r e of the problem i s presented s c h e m a t i c a l l y i n F i g u r e 3, with the stages represented by a p p r o p r i a t e l y numbered r e c t a n g l e s , and the arrows i n d i c a t i n g the i n p u t s and outputs t o the v a r i o u s stages. V R U U L B j ^ C j U ^ U B j Figure 3. MP1 as a 2-point boundary value problem. 41 3.3.2 MP2 As An I n i t i a l Value Problem For each input s t a t e of each stage of MP1, a management sequence a must be created such t h a t Eg. (3.3-7) i s maximized. T h i s means t h a t f o r each output s t a t e of each stage of MP 1, the i n n e r problem MP2 must be s o l v e d . The d e c i s i o n problem MP2 i s to f i n d a management sequence a^ = <a u l, a u T ) t c D € a p p l i e d to treatment u n i t u that w i l l o p t imize some management o b j e c t i v e d e f i n e d over the whole management u n i t U. The f o r m u l a t i o n of t h i s problem as a s e r i a l system i s most e a s i l y d e s c r i b e d a f t e r developing a simpler problem f i n d i n g a y t h a t o p t i m i z e s an o b j e c t i v e defined only over a s i n g l e treatment u n i t . The components of the s e r i a l system are d e s c r i b e d below: Stage: t t=1, ... , T Each stage t corresponds t o a d i s c r e t e time i n t e r v a l i n the time frame of the system, d e s c r i b e d i n s e c t i o n 3.1.2. D e c i s i o n v a r i a b l e s : a V i. e A • u t ^ - u t The d e c i s i o n v a r i a b l e i s the choice and i n t e n s i t y of a management a c t i o n to be a p p l i e d to u a t t. State v a r i a b l e s : v u t The i n p u t s t a t e of the s e r i a l system at stage t i s the management s t a t e d e s c r i b e d i n s e c t i o n 3.1.3 . The i n i t i a l value of the management s t a t e ( v u l ) i s known and d e f i n e d as the c u r r e n t management s t a t e of the treatment u n i t . 42 v , = v ° (3. 3-17) u l u i n t h i s f o r m u l a t i o n the f i n a l management s t a t e i s l e f t f r e e as a ch o i c e s t a t e v° , and may be chosen o p t i m a l l y . When MP2 i s to be optimized only with r e s p e c t t o a s i n g l e tr€atment u n i t , and not as part o f a management u n i t problem, n e i t h e r the i n i t i a l nor f i n a l values of the commodity s t a t e s need to be c o n s t r a i n e d and are ch o i c e s t a t e s C. , and c. , r e s p e c t i v l e y . Jul JUT State t r a n s i t i o n f u n c t i o n : M / i (3.3-18) v = M ( p , v , a ) u t u u u t u t The s t a t e t r a n s i t i o n f u n c t i o n i s simply the stand s i m u l a t i o n model Eg. (3.1-1). { v u , t + l ' C u , t + l } = M u ( { V u t ' C u t } ' a u t ) The r e l a t i o n s h i p of the output s t a t e v to the input u t s t a t e v at any stage i s given by the i n c i d e n c e i d e n t i t y (3.3-19) V = V u t u , t + l The s t a t e t r a n s i t i o n f u n c t i o n p r e d i c t s the treatment u n i t s t a t e at stage t+1 as a f u n c t i o n of the treatment u n i t s t a t e and management a c t i o n at stage t . Return f u n c t i o n : 43 R = R (v ,v ,,...,v ,,a ,,a . — , a ) u u uT u,T-l u l u l u 2 uT ^ 3 •3-20) T = Z R (v , a ) t = 1 ut ut ut The stage r e t u r n from s c h e d u l i n g a management a c t i o n a^t i s the a c t i o n r e t u r n d e s c r i b e d i n s e c t i o n 3.1.4. The t o t a l r e t u r n f o r a management sequence B^fa^) i s the sum of a l l the a c t i o n r e t u r n s i n c u r r e d i n the management seguence. The system of s t a g e s , s t a t e s and d e c i s i o n s d e s c r i b e d above can be c h a r a c t e r i z e d as a T - d e c i s i o n , one-state v a r i a b l e i n i t i a l v alue problem. The i n i t i a l s t a t e i s known and f i x e d , and T d e c i s i o n s must be made on the management a c t i o n i n each i n t e r v a l i n the time frame. The s e q u e n t i a l s t r u c t u r e of the s e r i a l system can be e x p l o i t e d to make the t o t a l manaqement sequence r e t u r n depend only cn the d e c i s i o n s and the i n i t i a l s t a t e v by using the t r a n s i t i o n f u n c t i o n Eg. (3.3-18) to e l i m i n a t e s t a t e s v through v uT R = R (v ,v m — ,v .,a ,,a „,...,a ) (3.3-21) u u uT u,T-l u l u l u 2 uT s u b s t i t u t i n g the t r a n s i t i o n f u n c t i o n H , R = R (M (...(M (v ,a ',P..),a _ ) , . . . , a _) ) u u ut u l u l u l u u 2 uT R (v ;a ,a ) u u l u l u 2 uT (3. 3-22) In summary, the i n i t i a l value maximum r e t u r n problem i s to f i n d the management a c t i o n s (a , a a ) f o r each treatment u l u 2 uT u n i t u, that o p t i m i z e the management seguence r e t u r n 44 * * * * R = R (v ;a ,a ,...,a ) u u u l u l u2 uT (3.3-23) such t h a t the i n i t i a l management s t a t e i s d e f i n e d to be the c u r r e n t management s t a t e of the treatment u n i t , v = v ° (3. 3-24) u l u l the e q u a l i t y c o n s t r a i n t s i m p l i e d by the t r a n s i t i o n f u n c t i o n s are met, v - M ( v , a , p ) = o u,t+l u ut ut u t = 1, . . . ,T (3. 3-25) and t h a t the management a c t i o n s are f e a s i b l e . a C A ut ut u = 1,...,U (3.3-26) The s e q u e n t i a l s t r u c t a r e of the problem i s presented s c h e m a t i c a l l y i n F i g u r e 4, with the stages r e p r e s e n t e d by a p p r o p r i a t e l y numbered r e c t a n g l e s , and the arrows i n d i c a t i n g the i n p u t s and outputs t o the v a r i o u s stages. n -•— Ru1 R u t = V l 1 t L V I T V 'u1 Figure 4. MP2 as an i n i t i a l value problem. 45 3.3.3 MF2 As A Subproblem Of MP 1 S o l u t i o n of the MP2 i n i t i a l value problem w i l l provide the optimum management seguence, a u , f o r each treatment u n i t * * * c o n s i d e r e d i n i s o l a t i o n from the eth e r s . I f aJL , a 2 , .. ., a y i s a f e a s i b l e candidate s e t f o r MP1, then i t i s a l s o o p t i m a l i n MP1. However f e a s i b i l i t y with regard t o the commodity c o n s t r a i n t s can only be guaranteed by i n c o r p o r a t i n g the commodity a l l o c a t i o n c o n s t r a i n t s d i r e c t l y i n t o MP2. T h i s i s accomplished through r e p r e s e n t i n g the t o t a l commodity input s t a t e at time i n t e r v a l t , i n the f o l l o w i n g manner. Sta t e v a r i a b l e : C j u t Let C j u t be the t o t a l amount of commodity j produced or consumed a f t e r treatment u n i t s 1, 2, u-1 have been scheduled, and treatment u n i t u has been scheduled up to time p e r i o d t - 1 . t - i C j u t = C j u ^ f ^ j u k j = l , . . . , J (3.3-27) Sta t e t r a n s i t i o n f u n c t i o n : C = w . r c . . , a . ) j u t j u t j u t u t = C. + c. (3.3-28) j u t j u t The s t a t e t r a n s i t i o n f u n c t i o n w . simply adds the amount J U . of commodity j produced or consumed at stage t as a 46 f u n c t i o n of the management a c t i o n a , to the running t o t a l . For stages 1, 0-1 of MP1, MP2 i s formulated as an i n i t i a l value problem, with the i n i t i a l t o t a l commodity s t a t e of MP2 d e f i n e d as the MP 1 stage u i n p u t s t a t e . By the MP1 i n c i d e n c e i d e n t i t i e s Eg. (3.3-6), C j u l = C u j = C j , u - 1 , T (3.3-29) At stage 0 of MP1, MP2 i s a 2 p o i n t boundary value problem with the f i n a l t o t a l commodity s t a t e d e f i n e d by the MP2 U stage output s t a t e : ~ = ~ Q (3.3-30) DUt j u The other elements of the MP2 d e c i s i o n problem remain unchanged. The combined MPT-MP2 s e r i a l system i s represented s c h e m a t i c a l l y i n F i g u r e 5. 47 c ? 1 5 , 5 c r J O a?1 rr n o >5 or 5 - 5 (0 ra o l U cr cr ra cr CJ o (0 cu u Cn •H Z d l A l 48 3.3.4 The S e r i a l M u l t i s t a g e Model The f o r e s t lands planning problem can now be expressed i n terms of the s e r i a l m u l t i s t a g e model: Find the D x T s e t of management a c t i o n s a such t h a t U T R = I E R (c ,v ,a ) (3.3-31) i , ut. u t u t u t u = l t = l i s o p t i m i z e d , and the commodity s t a t e t r a n s i t i o n c o n s t r a i n t s are met C. , - w. (C. ,a ) = 0 ,n -3 so\ j u , t + l j u t j u t u t (3.3-32) f o r a l l j , u , t , with u # 0, and t * T. The i n i t i a l commodity values C,, = c° (3. 3-33) n i l D can be d i r e c t l y s u b s t i t u t e d i n t o Eg. (3.3-32) but the f i n a l c o n d i t i o n s are represented e x p l i c i t l y : g.(c. ) = o j = l , . . . , J (3.3-34) The management s t a t e t r a n s i t i o n s must be met 49 V ~ M 4 - ( V 4 . ' A 4 . ) = 0 (3.3-35) u,t+l ut ut ut ut f o r a l l u t , t £ 1. The i n i t i a l management s t a t e v = v° (3.3-36) u l u u = 1,. .. ,U can be s u b s t i t u t e d i n t o Eg. (3.3-35). F i n a l l y , the management a c t i o n c o n s t r a i n t s must be met:' a £2. A u = i,...,u (3.3-37) ut ut t - 1 T 3.4 Approaches To O p t i m i z a t i o n 3.4.1 Decomposition By Dynamic Programming Dynamic programming e x p l o i t s the s e q u e n t i a l s t r u c t u r e of a s e r i a l system to transform the U x T - d e c i s i o n , J - s t a t e d e c i s i o n problem i n t o a set of u x T o n e - d e c i s i o n J - s t a t e problems. The theory of dynamic programming i s covered e x t e n s i v e l y elsewhere (Nemhauser, 1966 ; Wilde and B e i g h t l e r , 1967) but a s h o r t i n t u i t i v e review w i l l emphasize some of the assumptions of the technigue. A dynamic programming decomposition of the s e r i a l 50 m u l t i s t a g e model i n v o l v e s r e p l a c i n g the continuous s t a t e and s o l u t i o n space with d i s c r e t e values. Such d i s c r e t e v a r i a b l e problems can be represented as networks with the nodes a t every p o s s i b l e s t a t e and stage combination. For example, the d i s c r e t e v a r i a b l e f o r m u l a t i o n of the MP 1 s e r i a l m u l t i s t a g e problem would i n v o l v e nodes { ( u , t ) , c } u t where (u,t) i s the stage and C i s the value o f s t a t e v a r i a b l e s at t h a t stage. A l i n e a r model of the network would i n c l u d e a m a t e r i a l balance c o n s t r a i n t at each node. Dynamic programming e x p l o i t s the s e r i a l s t r u c t u r e of the system to s e q u e n t i a l l y e v a l u a t e the ut maximum r e t u r n f u n c t i o n . For MP1, the maximum re t u r n f u n c t i o n R i s de f i n e d as the stage 1 maximum r e t u r n f u n c t i o n , f , and i s c a l c u l a t e d with the usual dynamic programming r e c u r s i o n s . At the f i n a l stage 0, the output s t a t e s are a r b i t r a r i l y s e l e c t e d t o f u l f i l l the boundary c o n d i t i o n s LB. < C. < UB. D - DU - D For every f e a s i b l e combination of i n p u t and output s t a t e s , the d e c i s i o n i n v e r s i o n form of the t r a n s i t i o n f u n c t i o n r e t u r n s the a p p r o p r i a t e d e c i s i o n s et (management seguence) a^ to be used on treatment u n i t U. The D stage maximum r e t u r n i n v o l v e s no o p t i m i z a t i o n f (c ,c ) = Rfw' (C ,c ) ,c ) (.3.4-1) U u u u u u u Knowing the l a s t stage maximum r e t u r n and the t r a n s i t i o n f u n c t i o n W, one can determine the (U-1, 0-2, ... , 2) stage 51 r e t u r n s from the f o l l o w i n g r e c u r s i o n f (C ) = Maximum (R (C ,a ) + £ , (C ,a ) > (3.4-2) U U - _ *• u u u u+1 u u J a C A u u for u = U - l , . . . , 2 At the i n i t i a l s tage, the boundary c o n d i t i o n s d e f i n e the i n i t i a l s t a t e , f 1 ( B ) = Maximum { R^ (B,a ) + f 2 ( B , a 2 ) } (3.4-3) a i ^ A l The process i s repeated f o r v a r i o u s f e a s i b l e values of the f i n a l c o n d i t i o n s , as d i r e c t e d by an optimum search technique. Of course, computation of the above MP1 r e c u r s i o n s imply s o l u t i o n of the MP2 d e c i s i o n problem at each staqe u. When u = D, the boundary c o n d i t i o n C, = C, "1 — lr * f * f J JU.T. JU. must be met. The 0, T stage r e t u r n i s computed through d e c i s i o n i n v e r s i o n , s i m i l a r to Eg. (3.4-1). At each MP1 staqe u = U, MP2 i s an i n i t i a l value problem and can be s o l v e d d i r e c t l y , f . (C , v ) = Maximum {R (C , v , a . ) + f (C ^ _,v ,a )} ut ut U t _ ut ut ut ut u,t+l ut ut ut a t. A ut ut t = T , T - 1 , . . . , 2 u = U-l,... , 1 (3.4-4) 52 by means of the t r a n s i t i o n f u n c t i o n s H and W d e s c r i b e d i n u u s e c t i o n s 3.3.2 and 3.3.3. fit the i n i t i a l stage t = 1 the boundary c o n d i t i o n s d e s c r i b e the i n i t i a l s t a t e . f (C ,V ) = Maximum {R (C ,V ,a ) + f (C . ,V . ,a )} (3.4-5) u l u l u , . u l u l u l u l u2 u l u l u l a £ A u l u l u = 2,3,...,U-1 when u = 1 f l l ( B , V l l ) = M a x i m u m { R i i ( B ' v i i ' a n ) + f l 2 ( B , V l l ' a l l ) } (3.4-6) a l l ^ A l l Although the above r e c u r s i o n s t h e o r e t i c a l l y optimize the mu l t i s t a g e model of the timberlands management d e c i s i o n problem, the computational f e a s i b i l i t y o f the approach i s d o u b t f u l . For each value of every s t a t e v a r i a b l e , a dynamic program o p t i m i z a t i o n must be performed and the r e s u l t s s t o r e d . Hence dynamic programming i s u s u a l l y i m p r a c t i c a l when the dimensions of the s t a t e v a r i a b l e are g r e a t e r than 2 or 3, The c o n s t r a i n e d commodity s t a t e v a r i a b l e C has as many elements (J) as there are commodities c o n s t r a i n e d i n time p e r i o d s . I f volume pr o d u c t i o n i s c o n s t r a i n e d per decade over a plann i n g h o r i z o n of 20 decades, t h e r e are 20 elements i n C (J = 20). S i m i l a r l y , the management s t a t e may be m u l t i d i m e n s i o n a l . For example, i f Goulding's model (discussed i n Chapter 2) was used as the management s t a t e t r a n s i t i o n f u n c t i o n , the management s t a t e v would be the stand age and a DBH l i s t . 53 The replacement of n a t u r a l l y continuous s t a t e s and d e c i s i o n s with d i s c r e t e v a r i a b l e s a l s o causes computational problems. The r e s o l u t i o n of the d i s c r e t e v a r i a b l e f o r m u l a t i o n i n c r e a s e s with the number of d i s c r e t e c l a s s e s , but of course computational e f f o r t a l s o i n c r e a s e s . However, the most s e r i o u s problems concern the d e c i s i o n i n v e r s i o n of the management s t a t e t r a n s i t i o n f u n c t i o n . I f the t r a n s i t i o n f u n c t i o n i s simple enough i t can be sol v e d f o r the d e c i s i o n v a r i a b l e s i n terms of the input and output s t a t e . But i f the management s t a t e t r a n s i t i o n f u n c t i o n i s a stand or treatment u n i t model, d e c i s i o n i n v e r s i o n s may be impossible or not even meaningful. D e c i s i o n i n v e r s i o n may be circumvented with the i t e r a t i v e techniques of approximating Lagrange m u l t i p l i e r s d e s c r i b e d i n Appendix I I , but the computational e f f o r t r e g u i r e d i s c o n s i d e r a b l e . Most of the computational d i f f i c u l t i e s with the dynamic programming approach are r e l a t e d to the commodity c o n s t r a i n t s of MP1. The ME2 problem, independent of MP 1 and formulated as an i n i t i a l value problem, i s g u i t e t r a c t a b l e by dynamic programming, as w i l l be demonstrated i n l a t e r s e c t i o n s . 54 3.4.2 The D i s c r e t e Optimum P r i n c i p l e The Kuhn Tucker theorems s t a t e t h a t f o r a wide c l a s s of c o n s t r a i n e d programming problems, a Lagrangian e x p r e s s i o n can be formed that i n v o l v e s no c o n s t r a i n t s and i s g e n e r a l l y e a s i e r to o p t i m i z e . The l a g r a n g i a n has the u s e f u l property t h a t : whatever values of the v a r i a b l e s maximize (minimize) the value of the o r i g i n a l o b j e c t i v e f u n c t i o n s u b j e c t to i t s e q u a l i t y or i n e q u a l i t y c o n s t r a i n t s , w i l l maximize (minimize) the value cf the Lagrangian e x p r e s s i o n ( s u b j e c t o n l y to the n o n - n e g a t i v i t y c o n d i t i o n s f o r the v a r i a b l e s ) , Besides i t s u s e f u l n e s s as a computational d e v i c e , the Lagrangian approach o f f e r s a great deal of a n a l y t i c power. Here we w i l l use i t to i l l u m i n a t e the c o n d i t i o n s under which mu l t i s t a g e decomposition oc c u r s . The Lagrangian e x p r e s s i o n f o r the s e r i a l m ultistage model i s formed by a d j o i n i n g the c o n s t r a i n t s Eqs. (3.3.4-2) (3.3.4-4) to the o b j e c t i v e f u n c t i o n Eq. (3.3.4-1) by employing a J x nw x T matrix of m u l t i p l i e r s , X_ . i i + . and a 0 x T matrix a ^ . j L ( X , a ) = E E { R - E X . u t * U T - E X . 3 = 1 D g . i c . J U T 3 J U T - E E a ( v u t u , t + l [ ( v , c , a ) ) u t u t u t u t (3.4-7) This a n a l y s i s w i l l c o n c e n t r a t e on the Lagrange m u l t i p l i e r s X . corresponding to the c o n s t r a i n e d commodity t r a n s i t i o n 55 f u n c t i o n s , as i t i s the commodities t h a t are the common elements of the MP 1 and MP2 problems. Reference to the management s t a t e t r a n s i t i o n s w i l l be emitted t o s i m p l i f y the f o l l o w i n g d i s c u s s i o n . The d i s c r e t e optimum technigue e x p l o i t s the continuous nature of the Lagrangian e x p r e s s i o n to p r e d i c t (with a f i r s t o rder approximation) the improvement i n the o b j e c t i v e f u n c t i o n with chages i n management d e c i s i o n s . The d i s c r e t e optimum p r i n c i p l e e x p l o i t s the s e r i a l m u l t i s t a g e type of problem but i s more e f f i c i e n t than dynamic programming f o r systems with m u l t i d i m e n s i o n a l , continuous s t a t e s . Since most of the i n f o r m a t i o n generated by dynamic programming i s never used, i t would seem d e s i r a b l e to deal with s e l e c t e d values of the s t a t e s , r a t h e r than with a l l of them. The a l g o r i t h m s based on the maximum p r i n c i p l e s t a r t with a 'best guess 1 set of d e c i s i o n s , generate the v a l u e s of the s t a t e s r e s u l t i n g , and then decide how to a d j u s t the d e c i s i o n s so as to improve the o b j e c t i v e f u n c t i o n . T h i s p o l i c y improvement approach needs much l e s s computer storage although i t may r e q u i r e many t r i a l s i f the i n i t i a l guess i s bad or the f u n c t i o n i s poorly behaved. The d i s c r e t e optimum p r i n c i p l e has been widely and thoroughly d e s c r i b e d (Fan and Hang, 1964), and only a few elements of the approach w i l l be s t a t e d here. The d i s c r e t e optimum p r i n c i p l e i s a form of s t a t i n g the necessary c o n d i t i o n s f o r s o l u t i o n of m c l t i s t a g e d systems. To i n i t i a t e the d e s c r i p t i o n of the a l g o r i t h m , assume t h a t the matrix of Lagrange m u l t i p l i e r s i s a l r e a d y known. The 56 s t r a t e g y of the d i s c r e t e optimum approach i s t o c o n s t r u c t a problem f o r each stage such that i t s s o l u t i o n s a t i s f i e s the Kuhn-Tucker c o n d i t i o n s of the o r i g i n a l problem. T h i s e q u i v a l e n t problem i s to f i n d the s t a t i o n a r y p o i n t s o f a Hami l t c n i a n f u n c t i o n , with r e s p e c t to the management d e c i s i o n s . The ut stage Hamiltonian f o r the s e r i a l m ultistage model of f o r e s t l a n d planning i s : fi ^  = E E R. + I X. .. w. ( C . . ,a .) (3.4-8) ut , , ., km 3 ]u,t+l jut jut ut k=l_iH==l The p a r t i a l d e r i v a t i v e of the Hamiltonian with r e s p e c t to the d e c i s i o n i s the Kuhn-Tucker d s c i s i i o n d e r i v a t i v e : !!ut = ! V + EX. , + 1 ! ^ u t = «_£_ (3.4-9) ^ut 3aut * 9aut 6'aut When the Hamiltcnians are made s t a t i o n a r y by a d j u s t i n g the management d e c i s i o n s , the improved d e c i s i o n s are used to compute the new s t a t e s v i a the t r a n s i t i o n f u n c t i o n s . A new s e t of Lagrange m u l t i p l i e r s must then be obtained to i t e r a t e the process. When there i s no improvement i n the Hamiltonians, the process terminates. The problem of generating the a p p r o p r i a t e s e t s of Lagrange m u l t i p l i e r s w i l l now be examined. The Lagrangean e x p r e s s i o n Eg. (3.4-7) i s unconstrained so necessary c o n d i t i o n s f o r an i n t e r i o r optimum apply: f o r a l l u, t # UT 57 3 L 3 C . 3R ut j u t 9C j u t j u t J £ 1 = 1 3w. i u , t + l i u t (3. 4-10) 3 C j u t 3R ju t ut 3 C j u t J E i = l 3 w i u , t + l i u t 3 C . (3. 4-11) 'jut The r e c u r s i v e r e l a t i o n s h i p Eg. (3,4-11) r e l a t e s a l l the commodity j m u l t i p l i e r s to the unknown l a s t stage m u l t i p l i e r . At the l a s t stage, the boundary values impose the c o n d i t i o n x. JUT and g. (C. ) = 0 y j JUT X.r = 3 V + i X. , d g i (3.4-12) 3 C - T i - l 1 U ' T + 1 3 C 7 Note that i f the commodity I t r a n s i t i o n f u n c t i o n wj i s i u t independent of commodity j , then 3 w i u t = 0 i f i / j (3.4-13) 3 C ^ j u t as commodity c o n s t r a i n t s are u s u a l l y l i n e a r , 3 w . _ . . l u t = 1 i f i = D (3, n - 1 4 ) Furthermore, i f the commodity j i s not represented e x p l i c i t l y i n the o b j e c t i v e f u n c t i o n , 58 3 R u t = Q (3.4-15) 3 C . j u t I f these three assumptions are used to s i m p l i f y Eg. (3.4-16), a l l the commodity j m u l t i p l i e r s are i d e n t i c a l i n value and egual to the f i n a l stage m u l t i p l i e r X . = X . = ... = X. , (3.4-16) j u t j , u , t + l j u , T + l * Consequently, the najor problem i n a p p l y i n g a d i s c r e t e optimum p r i n c i p l e a l g o r i t h m to cur s e r i a l m u l t i s t a g e model, c o n s i s t s c f g e n e r a t i n g the a p p r o p r i a t e Lagrange m u l t i p l i e r s c o r r e s p o n d i n g to the f i n a l commodity s t a t e s . Dpon c o n s i d e r i n g computational s t r a t e g i e s , we encounter the same block as with dynamic programming. The two p o i n t boundary value problem . MP1 r e q u i r e s d e c i s i o n i n v e r s i o n to i n c o r p o r a t e the f i n a l c o n d i t i o n s i n t o a s o l u t i o n a l g o r i t h m . Again, as with dynamic programming, the search a l g o r i t h m d e s c r i b e d i n Appendix II can be used to circumvent d e c i s i o n i n v e r s i o n , but the computational e f f e c t i s c o n s i d e r a b l e . The d i s c r e t e optimum approach demonstrates how the Lagrange m u l t i p l i e r s can be used to l i n k the elements of a d e c i s i o n problem. The Lagrange m u l t i p l i e r s can be thought of as the •glue' which holds the problem t o g e t h e r . Each stage i s optimized s e p a r a t e l y , with the Lagrange m u l t i p l i e r s s e r v i n g as c o o r d i n a t i n g v a r i a b l e s between the i n d i v i d u a l o p t i m i z a t i o n s . With each adjustment of an MP2 d e c i s i o n v a r i a b l e , the commodity s t a t e v a r i a b l e s change i n value and a new set of m u l t i p l i e r s must be c r e a t e d . 59 3.4.3 Approximation With A L i n e a r Model The l i n e a r programming (LP) approach e s s e n t i a l l y s e l v e s MP 1 by assuming t h a t a candidate s e t , a, e x i s t s and i s complete, or a t l e a s t c o n t a i n s a l l the management seguences of i n t e r e s t . In forming the l i n e a r model, one a s s o c i a t e s with every management sequence a u £ L a# a column vec t o r of inp u t - o u t p u t c o e f f i c i e n t s t h a t r e p r e s e n t the c o n t r i b u t i o n s to each commodity c o n s t r a i n t equation of the a p p l i c a t i o n of the management sequence to one acre of land. Consider a l i n e a r form of the ge n e r a l commodity c o n s t r a i n t equation (introduced i n s e c t i o n 3.1.4) U T g. = v y c . = C. = C. (3.4-17) : I I j u t J U T j The assumption of an e q u a l i t y c o n s t r a i n t s i m p l i f i e s n o t a t i o n without l o s s of g e n e r a l i t y . Let -x be the number of acres of treatment u n i t u managed by the a l t e r n a t i v e management sequence ( k ) a . Then Eq. (3.4-17) can be r e w r i t t e n as U T 3c. g j " * I X u k = c j (3.4-18) u k A s s o c i a t e d with the M a l t e r n a t i v e management sequence f o r u 3C_ 3C. i s the vector of d e r i v a t i v e s 3 x . For e x a m p l e , — 3 — might be u k 3x , u k the volume/acre produced by c l e a r c u t t i n g treatment u n i t u i n the t h i r d decade. As with dynamic programming, a s h o r t d e s c r i p t i o n of the LP approach to o p t i m i z a t i o n w i l l help t o motivate the development 60 of a more s u i t a b l e o p t i m i z a t i o n a l g o r i t h m i n s e c t i o n 3.4,4. In terms of 8P1, the LP approach a l l o c a t e s acres of a treatment u n i t among a l t e r n a t i v e management sequences. Adjustment of x a f f e c t s the o b j e c t i v e f u n c t i o n B i n two ways. F i r s t , there i s a d i r e c t management sequence r e t u r n from a p p l y i n g a to u, 8R ax U K I n d i r e c t e f f e c t s cn R come from improvements in the management of the u n i t as a whole, through a l l o w i n g more e f f i c i e n t use of the c o n s t r a i n e d commodities. The i m p l i c i t value of a c o n s t r a i n e d commodity i s computed at each i t e r a t i o n of the LP process and i s represented as a d e r i v a t i v e , 6 R  6 - C J T h i s p r i c e i s a c o n s t r a i n e d d e r i v a t i v e ( a f t e r Wilde and B e i g h t l e r , 1967), meaning that the f u n c t i o n i s the r a t e of change of the o b j e c t i v e f u n c t i o n r e s u l t i n g from f e a s i b l e p e r t u r b a t i o n s i n the commodity s l a c k v a r i a b l e s . An example cf an i n f e a s i b l e p e r t u r b a t i o n would be i f the amounts of 'excess' of a commodity went n e g a t i v e . The t o t a l e f f e c t of a p e r t u r b a t i o n of x i s the u k c o n s t r a i n e d d e r i v a t i v e : 6R_ = 9R_ + z ^ -SB. (3.4-18) « x u k 8 X u k * 8 x u k 6 C j T h i s e x p r e s s i o n f o r the d e c i s i o n d e r i v a t i v e can be i n t e r p r e t e d as the unconstrained management sequence r e t u r n with a 61 c o n n e c t i o n f o r keeping the commodity c o n s t r a i n t s t i g h t . The IP s o l u t i o n a l g o r i t h m i t e r a t i v s l y computes new values f o r d e c i s i o n s v a r i a b l e s x • , and the a s s o c i a t e d c o n s t r a i n e d u k d e r i v a t i v e s , such t h a t complementary s l a c k n e s s i s maintained: 3 R x , =-o = 3 R 7 — u k — C . 3 x 3 C . j u k j J V - 1 v 3 ~ '' * •'J JS. — X / • • • / IS. When n o n - p o s i t i v i t y o f the c o n s t r a i n e d d e r i v a t i v e s i s obtained, | | - i 0 u = i,...,u (3. 4-20) u k k = 1 , . . . , K |*- - 0 j = 1 J (3, 4-21) 9C . D the optimal values have been obtained. L i n e a r programming e x p l o i t s the l i n e a r i t y of the o b j e c t i v e f u n c t i o n and c o n s t r a i n t e g u a t i o n s , and d i v i s i b i l i t y of the treatment u n i t s , to e f f i c i e n t l y a l l o c a t e the c o n s t r a i n e d commodities. Host of the commodity c o n s t r a i n t s i n the f o r e s t l a n d management plann i n g problem are e a s i l y expressed i n the form o f Eg. (3.4-17). Consequently LP has c l e a r advantages when commodities are c o n s t r a i n e d a c r o s s the whole management u n i t . Furthermore, l a r g e g e n e r a l i z e d LP codes are a v a i l a b l e to handle the very l a r g e problem s i z e a s s o c i a t e d with f o r e s t p l anning. In a d d i t i o n to s o l u t i o n a l g o r i t h m s , these computer codes o f f e r a s e l e c t i o n of post optimal procedures to a s s i s t i n 62 an a l y s e s of the planning model. The most s u c c e s s f u l f o r e s t p lanning models, Timber RAM (Navon, 197 0) and Maxmillion ( C l u t t e r et a l , 1S69) c a p i t a l i z e on these f e a t u r e s of l i n e a r programming. The disadvantages of the LP approach d e r i v e from the f a c t t h a t the l i n e a r model i s a l e s s a c curate approximation of the r e a l planning problem, than the s e r i a l m u l t i s t a g e model. Although the commodity c o n s t r a i n t s are r e a d i l y expressed as l i n e a r e g u a t i o n s , the l i m i t a t i o n s of a l i n e a r o b j e c t i v e f u n c t i o n i s much l e s s s a t i s f a c t o r y . C o n s t r u c t i o n of the l i n e a r model to s o l v e MP1 and MP2 si m u l t a n e o u s l y assumes t h a t the complete s e t of a l t e r n a t i v e management sequences necessary to compute the optimum, i s known a p r i o r i . Yet, i f a d e c i s i o n v a r i a b l e was c r e a t e d t o rep r e s e n t a l l f e a s i b l e management sequences f o r a r e a l i s t i c p l anning problem, s o l u t i o n would be imp o s s i b l e due to the very l a r g e number of v a r i a b l e s , A treatment u n i t with more than 28,000 a l t e r n a t i v e management sequences w i l l be d e s c r i b e d i n l a t e r s e c t i o n s . U s u a l l y only a s m a l l number of p o s s i b l e management sequences are e x p l i c i t l y represented i n the l i n e a r model. In summary, the LP approach c a p i t a l i z e s on the l i n e a r nature of the c o n s t r a i n t equation and provides an e f f i c i e n t s o l u t i o n to MP1, The MP2 problem i s avoided completely by r e g u i r i n q the candidate management sequences to be known a p r i o r i and i n c l u d e d i n the l i n e a r model. 6 3 3.4.4 Dantzig-Wolfe Decomposition In s e c t i o n s 3.4.1 and 3.4.2 »€ have seen t h a t the MP2 problem s c h e d u l i n g management a c t i o n s on treatment u n i t s i s best accomplished through a l g o r i t h m s that e x p l o i t the problem's s e r i a l m u l t i s t a g e s t r u c t u r e . However, the commodity a l l o c a t i o n problems (HF1), i n which the s c h e d u l i n g problems are embedded, i s c o m p u t a t i o n a l l y i n f e a s i b l e to optimize v i a the usual dynamic programming r e c u r s i o n s , due to a continuous and m u l t i d i m e n s i o n a l s t a t e space. The d i s c r e t e optimum approach provides an improvement i n computational e f f i c i e n c y f o r the continuous, m u l t i d i m e n s i o n a l commodity s t a t e space but, as with dynamic programming, i t r e q u i r e s d e c i s i o n i n v e r s i o n c f the t r a n s i t i o n f u n c t i o n s to handle the boundary values of the commodity s t a t e s . Where the s t a t e t r a n s i t i o n f u n c t i o n i s a reasonably r e a l i s t i c f o r e s t lands s i m u l a t o r , d e c i s i o n i n v e r s i o n i s u n l i k e l y to be f e a s i b l e . Conversely, approximation of MP1 with a l i n e a r model and s o l u t i o n of the commodity a l l o c a t i o n problem with l i n e a r programming, i s very e f f i c i e n t . T h i s approach i g n o r e s the s e r i a l staged nature of the problem to e x p l o i t the l i n e a r form c f the commodity c o n s t r a i n t s . Consequently, the n a t u r a l l y time staged and h i g h l y n o n - l i n e a r KP2 problem cannot be handled d i r e c t l y . The LP approach assumes a candidate set i s s e l e c t e d f o r commodity a l l o c a t i o n . I f the candidate s e t c o n t a i n s every p o s s i b l e management seguence, MP2 i s solved through exhaustive enumeration. ft s y n t h e s i s of these two approaches i s suggested by the 64 r o l e of the Lagrange m u l t i p l i e r s i n the d i s c r e t e optimum s o l u t i o n s t r a t e g y . The m u l t i p l i e r s serve as c o o r d i n a t i n g v a r i a b l e s between the u" x T s t a g e s , a l l o w i n g them to be o p t i m i z e d i n d i v i d u a l l y . By e x t e n s i o n , each MP2 problem can be o p t i m i z e d i n d i v i d u a l l y , while 'glued* to the other MP2 problems with the Lagrange m u l t i p l i e r s . The key t o combining the two approaches l i e s i n the i n t e r p r e t a t i o n of the Lagrange m u l t i p l i e r s a s s o c i a t e d with the l a s t stage commodity output s t a t e * t + i . T h i s m u l t i p l i e r i s the c o n s t r a i n e d d e r i v a t i v e of the o b j e c t i v e f u n c t i o n B with r e s p e c t to the f i n a l c o n d i t i o n c o n s t r a i n t . X = ££_ = (3.4-22) jU,T+l 6c. 6g. 1 j u t i A more i n t u i t i v e i n t e r p r e t a t i o n of the m u l t i p l i e r i s the i nstantaneous value to the o b j e c t i v e f u n c t i o n of another u n i t of the r i g h t hand s i d e , c . JUT The LP s o l u t i o n of the l i n e a r approximation of MP1 p r o v i d e s these commodity p r i c e s a u t o m a t i c a l l y . Although the MP 1 problem i s the optimal a l l o c a t i o n of commodities, the dual of t h i s problem i s to set e f f i c i e n t commodity p r i c e s . The dual v a r i a b l e s are the c o n s t r a i n e d d e r i v a t e s of the o b j e c t i v e f u n c t i o n B with r e s p e c t to the commodity c o n s t r a i n t . The i d e n t i f i c a t i o n of the Lagrange m u l t i p l i e r of the m u l t i s t a g e model as the dual v a r i a b l e of the l i n e a r model suggests the f o l l o w i n g a l g o r i t h m : 1 : An i n i t i a l f e a s i b l e s o l u t i o n a f i s provided f o r each MP2 u problem. 65 2 : The MF2 d e c i s i o n s are imposed on a l i n e a r model cf MP 1. 6 R LP o p e r a t i o n s provide a vector of dual v a r i a b l e s T — , i 3 : Using the i d e n t i t y Eg. (3.4-22), the stage ut Lagrange m u l t i p l i e r s are computed r e c u r s i v e l y v i a Eg. (3.4-11). 4 : The Hamiltonians Eg. (3.4-8) can then be w r i t t e n and optimized with r e s p e c t to the management d e c i s i o n s , e i t h e r stage by stage or a l l a t once. 5 : I f improvement i n the Hamiltonians i n st e p 4 was accomplished i n step 4, the management seguences are added to the LP ca n d i d a t e s e t , and the process i s repeated from step 2 . 6 : I f the improvement of the Hamiltonians i n st e p 4 was i n s i g n i f i c a n t , the process terminates. At t e r m i n a t i o n , the management seguences i n the LP b a s i s comprise the optimal schedule cf management a c t i o n s f o r the f o r e s t lands planning problem. The Hamiltonian i s a c o n s t r u c t c l o s e l y r e l a t e d to the Kuhn-Tucker d e f i n i t i o n o f the c o n s t r a i n e d d e c i s i o n d e r i v a t i v e Eg. (3.4-9), but e x p l o i t s the multistage nature of the problem. A more g e n e r a l approach at step 4 would be t c c o n s t r u c t a problem t h a t maximizes the d e c i s i o n d e r i v a t i v e d i r e c t l y . For any s e t of MP1 dual commodity p r i c e s , we can c o n s t r u c t a problem t h a t o p t i m i z e s MF2, and f u l f i l l s the MP1 Kuhn-Tucker c o n d i t i o n of n o n - p o s i t i v i t y (maximization) of the d e c i s i o n d e r i v a t i v e i f MP1 i s optimized. T h i s i s accomplished by using the MP1 d e c i s i o n d e r i v a t i v e as the o b j e c t i v e f u n c t i o n of the MP2 problem. 66 For each treatment u n i t u, f i n d the management sequence a u = { a u l ' a u 2 ' a u T J £ flT' S U C h t h a t 6 x 3 x , j S C . ( J . 4 - 2 J ) u k u k J 3 u k i s a maximum, and t h a t the management s t a t e t r a n s i t i o n s Eg. (3. 3. 4-4) are met v „ f x i - M ( v , c , a ) = 0 t = 1 , . . . , T u , t + l u t u t u t u t The d e c i s i o n d e r i v a t i v e form of MP2 r e p l a c e s steps 3 and 4 above. Termination of the process occurs when the maximum value of the o b j e c t i v e f u n c t i o n i s zero or negative. When |f 1 0 (3.4-24) u k f o r a l l u, the Kuhn-Tucker c o n d i t i o n of n o n - p o s i t i v i t y of the d e c i s i o n d e r i v a t i v e i s met f o r MP1. As complementary s l a c k n e s s i s maintained f o r HP 1 by the LP a l g o r i t h m and c o n v e x i t y can be assumed, MP 1 i s o p t i m a l . This technique o f using the dual v a r i a b l e s to c o n s t r u c t a new d e c i s i o n v e c t o r at each i t e r a t i o n o f an LP i s known as Wolfe-Bantzig decomposition (Bantzig and Wolfe, 1961). In the next chapter, some a l t e r n a t i v e forms of the MP2 subproblem w i l l be presented and optimized. In chapter 5 these subproblems w i l l be combined with a Timber RAM f o r m u l a t i o n of MP1, v i a Bantzig-Wolfe decomposition. 67 4* Q p t i f i z a t i p n Of The Sufcprcblem The mathematical s t r u c t u r e of the problem of f i n d i n g the o p t i m a l seguence of management a c t i o n s f o r a f o r e s t l a n d u n i t has been examined i n s e c t i o n s 3.3 and 3.4. I f the model c f the l a n d u n i t c o n s i s t s of w e l l behaved, a n a l y t i c f u n c t i o n s , the t echniques of s e c t i o n s 3.3.1 and 3.3.2 c o u l d be used d i r e c t l y . The o b j e c t i v e of t h i s chapter i s to d e s c r i b e o p t i m i z a t i o n techniques t h a t can be a p p l i e d to c u r r e n t l y a v a i l a b l e f o r e s t stand models, which are not g e n e r a l l y w e l l behaved i n the mathematical sense, but have earned acceptance as v a l i d management t o o l s . Two methods w i l l be d e s c r i b e d . F i r s t , a technigue f o r o p t i m i z i n g stand models d i r e c t l y w i l l be developed. The second method uses the stand model embedded i n an e f f i c i e n t dynamic programming f o r m u l a t i o n . 4.1 D i r e c t O p t i m i z a t i o n Shen an a n a l y t i c e x p r e s s i o n f o r the o b j e c t i v e f u n c t i o n i s e i t h e r u n a v a i l a b l e or too complicated to manipulate by the i n d i r e c t methods c f s e c t i o n s 3.3.1 and 3.3.2, there are two approaches to o p t i m i z a t i o n ; approximation technigues and d i r e c t methods. Approximation technigues e v a l u a t e the o b j e c t i v e at many p o i n t s i n order to approximate i t by an e x p r e s s i o n amenable to i n d i r e c t methods. As a somewhat s i m p l i s t i c example, Goulding (1972) f i t t e d the response s u r f a c e of volume y i e l d over a range 68 of values of d e c i s i o n v a r i a b l e s , such as i n i t i a l d e n s i t y and age a t h a r v e s t . As the d e c i s i o n space was two d i m e n s i o n a l , the response s u r f a c e was p l o t t e d , and the ' o p t i m i z a t i o n ' problem could be s o l v e d by i n s p e c t i o n . D i r e c t methods s t a r t at an a r b i t r a r y p o i n t and proceed stepwise toward the optimum by s u c c e s s i v e improvements. Any d i r e c t o p t i m i z a t i o n scheme using past i n f o r m a t i o n to generate b e t t e r p o i n t s i s c a l l e d a c l i m b i n g procedure. The a p p l i c a b i l i t y o f dirjc.t, c l i m b i n g technigues t o stand s i m u l a t i o n models w i l l be c o n s i d e r e d here. A general treatment of a wider c l a s s of optimum seek i n g t e c h n i q u e s , i n c l u d i n g d i r e c t c l i m b i n g , can be found i n Wilde (1964). D i r e c t c l i m b i n g techniques can be f u r t h e r s u b d i v i d e d i n t o g r a d i e n t methods and p a t t e r n searches. The g r a d i e n t method, going back to Cauchy, uses the plane tangent to the response s u r f a c e to i n d i c a t e the d i r e c t i o n of improvement of a p o l i c y . At each new point on the response s u r f a c e the g r a d i e n t i s approximated, and the p o l i c y i s improved along the l i n e of s t e e p e s t ascent. The step s i z e along the g r a d i e n t d i r e c t i o n i s found by a search technigue or, when p o s s i b l e , d i r e c t d i f f e r e n t i a t i o n . The l a t t e r i n v o l v e s s u b s t i t u t i n g the eguation of the l i n e of s t e e p e s t ascent i n t o the o b j e c t i v e f u n c t i o n , and maximizing t h i s e x p r e s s i o n f o r the l e n g t h of the l i n e using c l a s s i c a l c a l c u l u s techniques. At the new p o i n t , a new g r a d i e n t i s e valuated and the process i s repeated. The most e f f e c t i v e g r a d i e n t technigues have e f f i c i e n t r i d g e f o l l o w i n g p r o p e r t i e s . In a d d i t i o n , two important technigues, the p a r a l l e l tangents method of Shah, Buehler and Kempthcrne (1961) 69 and the d e f l e c t e d g r a d i e n t procedure c f F l e t c h e r and Powell (1963) have i d e a l behaviour cn q u a d r a t i c o b j e c t i v e s . The d e f l e c t e d q r a d i e n t approach has the p r o p e r t y of g u a d r a t i c convergence and i s g e n e r a l l y considered the best choice (Wilde and B e i g h t l e r , 1967). However, the procedure r e q u i r e s the f i r s t d e r i v a t i v e of the o b j e c t i v e f u n c t i o n with r e s p e c t t o the d e c i s i o n v a r i a b l e s , e i t h e r i n a n a l y t i c a l form or estimated n u m e r i c a l l y (Powell, 1964). S i m i l a r l y , the g r a d i e n t p a r a l l e l tangent (partan) search, with q u a s i q u a d r a t i c convergence, r e q u i r e s that the f i r s t d e r i v a t i v e s be s u p p l i e d or estimated. One advantaqe of a l l q r a d i e n t methods i s t h a t they i n h e r e n t l y a v o i d saddle p o i n t s . While the above g r a d i e n t techniques are e f f i c i e n t on g u a d r a t i c s u r f a c e s , they become much l e s s e f f i c i e n t as the o b j e c t i v e f u n c t i o n departs from the q u a d r a t i c i d e a l . To improve the e f f i c i e n c y cf o p t i m i z a t i o n , Buehler et a l . (1961) recommended that o p t i m i z a t i o n problems be transformed where p o s s i b l e : remove i n t e r a c t i o n between independent v a r i a b l e s , make the contours as s p h e r i c a l as p o s s i b l e by symmetric c h o i c e of s c a l e s of measurement, and r e p r e s e n t the o b j e c t i v e f u n c t i o n by a f u n c t i o n w e l l approximated by a low order T a y l o r expansion i n the neighborhood c f the optimum. Although the a c c e l e r a t e d and d e f l e c t e d g r a d i e n t a l g o r i t h m s developed s i n c e the Buehler et a l . (1961) r e p o r t have g r e a t l y improved performance cn non-guadratic s u r f a c e s , g r a d i e n t methods are s t i l l g e n e r a l l y s e n s i t i v e to problems of s c a l i n g and r e p r e s e n t a t i o n . Consequently, the more robust p a t t e r n search approach was chosen t c o p t i m i z e the complicated o b j e c t i v e s u r f a c e of the f o r e s t stand model. 70 P a t t e r n searches attempt t o use i n f o r m a t i o n gathered at v a r i o u s stages throughout the search f o r the optimum. Conseguently they are l e s s s e n s i t i v e t o l o c a l i r r e g u l a r i t i e s i n the o b j e c t i v e s u r f a c e . F i r s t or second d e r i v a t i v e s are not r e q u i r e d . As with g r a d i e n t techniques, p a t t e r n searches f i n d r i d g e s and f e l l o w them to the neiqhborhocd of the optimum using a c c e l e r a t i o n methods. A search a l g o r i t h m t h a t i s r e p r e s e n t a t i v e of the approach i s presented i n d e t a i l i n the next s e c t i o n . 4.1.1 The S e g u e n t i a l Simplex Algorithm Spendley, Hext and Himsworth (1962) i n t r o d u c e d a simple but i n g e n i o u s idea f o r adaptive c o n t r o l through e v a l u a t i n g the output from a system at a set of p o i n t s forming a simplex i n the f a c t o r space, and c o n t i n u a l l y forming new s i m p l i c e s by r e f l e c t i n g one p o i n t i n the hyperplane of the remaining p o i n t s . A simplex i s the N-dimensional g e n e r a l i z a t i o n of the e q u i l a t e r a l t r i a n g l e (N=2) and the r e g u l a r t e t r a h e d r o n (N=3). Nelder and Mead (1965) improved the a l g o r i t h m f o r o p t i m i z a t i o n a p p l i c a t i o n s by causing the simplex to adapt i t s e l f to the l o c a l landscape, e l o n g a t i n g down long i n c l i n e d v a l l e y s (or r i d g e s ) , changing d i r e c t i o n on encountering a v a l l e y (ridge) at an angle, and c o n t r a c t i n g at an optima. A b r i e f d e s c r i p t i o n o f the a l g o r i t h m f o r maximization f e l l o w s : 1 : Create the v e r t i c e s o f the i n i t i a l simplex by e v a l u a t i n g the o b j e c t i v e f u n c t i o n at n+1 p o i n t s i n the space of the 71 n independent v a r i a b l e s . 2 : The vertex at which the f u n c t i o n takes the s m a l l e s t value (Xmin) i s determined, and i s o v e r - r e f l e c t e d through the c e n t r o i d of the remaining v e r t i c e s (X) to gi v e a t r i a l p o i n t (Xnew) by the r e l a t i o n Xnew = (1 + a ) - x - a (Xmin) where a >1 i s the r e f l e c t i o n c o e f f i c i e n t . 3 : I f the t r i a l p o i n t proves to be a new maximum, go to step H. I f the t r i a l point proves s t i l l to be the worst p o i n t , go to step 5. Otherwise, r e p l a c e Xmin by Xnew and go t o step 2. 4 : Attempt an expansion d e f i n e d by the r e l a t i o n Xexp = u(Xnew) + (1- u> ) X where u >1 i s the expansion c o e f f i c i e n t . S e l e c t the best p o i n t of Xnew and Xexp to r e p l a c e Xmin i n the simplex, and go to step 2. 5 : Attempt a c o n t r a c t i o n by the r e l a t i o n Xcon = 3 Xmin + (1-3 )X where 0<3 <1 i s the c o n t r a c t i o n c o e f f i c i e n t . I f Xcon i s b e t t e r than Xmin, r e p l a c e Xmin i n the simplex with Xcon and r e t u r n to step 2. I f Xcon does not improve the simplex, the c o n t r a c t i o n has f a i l e d and a l l the v e r t i c e s (X ) of the simplex are moved c l o s e r to the best point (Xmax) by the r e l a t i o n X = (X + Xmax)/2 before going t o step 2. A r e f l e c t i o n , expansion and c o n t r a c t i o n o p e r a t i o n are represented i n F i g u r e 6. contraction F i g u r e 6. E l e m e n t s o f a s i m p l i c i a l s e a r c h : r e l e c t i o n , e x p a n s i o n a n d c o n t r a c t i o n . 73 The s e q u e n t i a l simplex method was chosen as the o p t i m i z a t i o n a l g o r i t h m to be t e s t e d with f o r e s t stand models due to a number of f a v o r a b l e c h a r a c t e r i s t i c s . The simplex search does not r e g u i r e f i r s t or second d e r i v a t i v e s , performing o n l y f u n c t i o n (model) e v a l u a t i o n s ( s i m u l a t i o n s ) . T h i s i s a p r e r e g u i s i t e f o r an a l g o r i t h m s u i t a b l e f o r o p t i m i z i n g s i m u l a t i o n models, as i t i s u s u a l l y i m p o s s i b l e t o formulate the d e c i s i o n d e r i v a t i v e s of any but the s i m p l e s t model. As a p a t t e r n search procedure, i t uses past i n f o r m a t i o n to generate an improved p o l i c y . Information about the landscape o f the o b j e c t i v e f u n c t i o n i s coded i n the shape of the adapting simplex. The step s i z e v a r i e s with the landscape, f i n d i n g and a c c e l e r a t i n g along r i d g e s , and s h r i n k i n g near optima. The method i s c o m p u t a t i o n a l l y compact, and economical as i t r e g u i r e s only one a d d i t i o n a l f u n c t i o n (model) e v a l u a t i o n (simulation) per i t e r a t i o n . The a l g o r i t h m , as i t i s presented above, was implemented with few m o d i f i c a t i o n s . The a d a p t a t i o n c o e f f i c i e n t s (a , u , 3 ) were chosen to be (1, 2, 1/2) as recommended by Nelder and Mead i n the o r i g i n a l paper. The r o u t i n e stops i t e r a t i n g i f the change i n the o b j e c t i v e f u n c t i o n i s l e s s than .001, or when the maximum number of i t e r a t i o n s s p e c i f i e d by the OPTIMIZE command has been reached. The t e r m i n a l user can i n t e r r u p t the o p t i m i z a t i o n procedure at any time by generating an ATTENTION i n t e r r u p t . One d e v i a t i o n from the o r i g i n a l a l g o r i t h m i s the a b i l i t y to handle ' e x p l i c i t ' or simple upper and lower bound c o n s t r a i n t s . I f a simplex o p e r a t i o n p r o j e c t s a t r i a l p o i n t that v i o l a t e s one or more e x p l i c i t c o n s t r a i n t s , the t r i a l p o i n t i s moved to the border of f e a s i b l e r e g i o n by a d j u s t i n g the bounded v a r i a b l e . 74 • I m p l i c i t ' c o n s t r a i n t s , which are f u n c t i o n s of more than one d e c i s i o n v a r i a b l e , could be accommodated by a simplex o p e r a t i o n recommended by Box (1965) , that would move a t r i a l p o i n t halfway towards the c e n t r c i d i f i t v i o l a t e d an i m p l i c i t c o n s t r a i n t . T h i s f a c i l i t y was not i n c l u d e d i n the o p t i m i z i n g r o u t i n e . M o d i f i c a t i o n s were considered to accommodate s t o c h a s t i c models. Assuming t h a t the varia n c e of the performance c r i t e r i o n i s homogenous over i t s range, a procedure was developed using a s t a t i s t i c a l t e s t to decide whether two p o l i c y p o i n t s had s i g n i f i c a n t l y d i f f e r e n t o b j e c t i v e v a l u e s . T h i s procedure was not implemented i n the o p t i m i z i n g s u b r o u t i n e , but the a l g o r i t h m i s d e s c r i b e d i n Appendix I I I . I n s t e a d , the a b i l i t y was provided t o s p e c i f y a number of independent r e p e t i t i o n s of e v a l u a t i o n s a t each p o l i c y p o i n t and the o b j e c t i v e value i s c a l c u l a t e d as an average. This f a c i l i t y , along with the i n t e r a c t i v e nature of the s u p e r v i s o r , allows the user to s p e c i f y more accuracy as the optimal p o l i c y i s approached. 4.1.2 The O p t i m i z a t i o n S u p e r v i s o r Program A s i m u l a t i o n o p t i m i z a t i o n program was w r i t t e n with the o b j e c t i v e of p r o v i d i n g a s u p e r v i s o r f o r a p p l y i n g d i r e c t c l i m b i n g o p t i m i z a t i o n a l g o r i t h m s t o v a r i o u s s i m u l a t i o n models. The system (Figure 7) was designed to be h i g h l y i n t e r a c t i v e as i t was expected t h a t the n o n l i n e a r and d i s c o n t i n u o u s nature of the e x i s t i n g f o r e s t land management models would r e g u i r e frequent 7 5 Commands READ D I S P L A Y E D I T S E T SIMULATE F I X F R E E O P T I M I Z E SUBROUTINE WRITE EOF STOP C o n v e r s a - \ t i o n a l y T e r m i n a l / U s e r S e s s i o n h a r d c o p y a n d r e p o r t s O p t i m i z a t i o n S u p e r v i s o r o p t t i p i i z e S e q u e n t i a l S i m p l e x A l g o r i t h m G o u l d i n g " ! M e y e r s > K i l k k i J S t a n d M o d e l L i b r a r y F i g u r e 7 . T h e S i m u l a t i o n O p t i m i z a t i o n (SIMOPT) s u p e r v i s o r s y s t e m . 7 6 i n t e r v e n t i o n f o r human d e c i s i o n making i n the course of an o p t i m i z a t i o n a n a l y s i s . The s u p e r v i s o r i s w r i t t e n i n FORTRAN but i s h e a v i l y dependent on MTS 7 u t i l i t i e s f o r in p u t / o u t p u t and c h a r a c t e r h a n d l i n g . The s u p e r v i s o r code i s l i s t e d i n Appendix IV, The s u p e r v i s o r c u r r e n t l y r e c o g n i z e s twelve commands, which are b r i e f l y d e s c r i b e d below. READ The READ command causes the s u p e r v i s o r to lead a p o l i c y with a s p e c i f i e d i d e n t i f i e r l a b e l from a given l o g i c a l u n i t . The f i l e POLICY i s assigned to a l o g i c a l u n i t when the program i s f i r s t s t a r t e d . Each p o l i c y on the POLICY f i l e s p e c i f i e s a s i m u l a t i o n model, i t s parameters and v a r i a b l e s , and a d d i t i o n a l i n f o r m a t i o n needed f o r the o p t i m i z a t i o n procedure. The POLICY f i l e w i l l be more completely d e s c r i b e d i n the t e s t problems of a l a t e r s e c t i o n . DISPLAY The DISPLAY command causes the c u r r e n t p o l i c y parameters and v a r i a b l e s to be d i s p l a y e d with t h e i r corresponding i n t e r n a l i n t e g e r i d e n t i f i e r s . 7 A l l development work was done on an IBM 370-168 o p e r a t i n g under the Michigan Terminal System (MTS). 77 EDIT The EDIT command causes t r a n s f e r of c o n t r o l to the MTS e d i t o r system and i n i t i a t e s the e d i t i n g of the c u r r e n t p o l i c y f i l e . A l l the f a c i l i t i e s of the MTS e d i t o r are a v a i l a b l e . At the end of the e d i t , the s u p e r v i s o r can be r e s t a r t e d without r e - i n i t i a l i z i n g the system. SET The SET command allows the user to a s s i g n values to the c u r r e n t p o l i c y v a r i a b l e s or parameters. SIMULATE The SIMULATE command causes the s u p e r v i s o r to pass the c u r r e n t p o l i c y to the a p p r o p r i a t e s i m u l a t i o n model. On e x e c u t i o n , t a b u l a r r e s u l t s may be d i s p l a y e d at the t e r m i n a l or d i r e c t e d t o a s c r a t c h f i l e f o r l a t e r p r i n t i n g . A r e p e t i t i o n f a c t o r allows the c a l c u l a t i o n of average r e t u r n s f o r s t o c h a s t i c models. F I X The .FIX command causes a v a r i a b l e to hold i t s c u r r e n t value throughout the o p t i m i z a t i o n process. FREE The FREE command r e l e a s e s a v a r i a b l e to take cn i t s optimium value during the o p t i m i z a t i o n process. 78 OPTIMIZE The OPTIMIZE command causes the t r a n s f e r of c o n t r o l to an o p t i m i z a t i o n a l g o r i t h m f o r a s p e c i f i e d number of i t e r a t i o n s . The values of the o b j e c t i v e f u n c t i o n and the f r e e v a r i a b l e s are d i s p l a y e d at each i t e r a t i o n and the user can i n t e r v e n e i f the a l g o r i t m s t a l l s or wanders out of the r e g i o n of i n t e r e s t . A r e p e t i t i o n f a c t o r allows o p t i m i z a t i o n based on average values f o r s t o c h a s t i c models. The improved p o l i c y r e p l a c e s the c u r r e n t p o l i c y at each i t e r a t i o n . SUBROUTINE The SUBROUTINE command causes the s u p e r v i s o r t o pass c o n t r o l to a user s u p p l i e d s u b r o u t i n e . WRITE The WRITE command causes the c u r r e n t p o l i c y to be saved on a s p e c i f i e d l o g i c a l u n i t . T h i s a l l o w s the user to save an optimized p o l i c y f o r f u r t h e r p r o c e s s i n g . EOF An end of f i l e s i g n a l generated i n response to a s u p e r v i s o r i n p u t prompt w i l l cause t r a n s f e r of c o n t r o l to the o p e r a t i n g system. Under MTS , p o l i c y s c r a t c h f i l e s can be saved on permanent f i l e s , and s i m u l a t i o n program r e p o r t s copied to the l i n e p r i n t e r . The s u p e r v i s o r i s 79 re-entered without r e - i n i t i a l i z i n g . STOP The STOP command causes a normal e x i t from the s u p e r v i s o r to MTS, with no r e s t a r t c a p a b i l i t y . Acronyms, e s p e c i a l l y i n academic s i t u a t i o n s , are unavoidably p r e t e n t i o u s . However, t h e i r use a l l o w s c l a r i t y and economy i n t e c h n i c a l w r i t i n g . The s i m u l a t i o n o p t i m i z a t i o n s u p e r v i s o r w i l l p e r i o d i c a l l y be r e f e r r e d to as SIMOFT. 4.1.3 Test Case: Meyer's Model Meyers* (1971) e m p i r i c a l , whole s t a n d / d i s t a n c e independent model has had e x t e n s i v e o p e r a t i o n a l use by the United S t a t e s F o r e s t S e r v i c e , In t h i s s e c t i o n , a problem c f o p t i m i z a t i o n a n a l y s i s w i l l be performed on Meyers' model with the i n t e r a c t i v e s u p e r v i s o r . T h i s e x e r c i s e w i l l demonstrate the f a c i l i t i e s of the s u p e r v i s o r system and help to e v a l u a t e the u t i l i t y of Meyers' approach i n an o p t i m i z a t i o n a n a l y s i s . A b r i e f o u t l i n e of Meyers' a l g o r i t h m i s necessary. The s t a t e v a r i a b l e s of Meyers' model are stand dbh, stand b a s a l area per acre and stand age, and the core of the model i s an e m p i r i c a l r e l a t i o n s h i p between two of these v a r i a b l e s , dbh and b a s a l area. R e s u l t s c f t h i n n i n g s t u d i e s are used t o generate a f a m i l y cf curves r e l a t i n g b a s a l area a f t e r t h i n n i n g to average stand diameter f o r standard l e v e l s of growing stock. The 80 •growing stock l e v e l ' (GSL) d e s i g n a t i o n f o r each curve i s i t s unique b a s a l area at an average diameter of 10.0 i n c h e s . Meyers' model s t a r t s with a d e s c r i p t i o n of the young stand p r i o r to t h i n n i n g : stand d e n s i t y , average diameter, s i t e index and age. The average height of dominants and codominants i s computed from r e g r e s s i o n f u n c t i o n s of s i t e index and age. fi p a r t i a l cut to a s p e c i f i e d growing stock l e v e l i s simulated by computing the change i n diameter and height due to t h i n n i n g , computing thinned stand volume, and c a l c u l a t i n g the t h i n n i n g volume as the d i f f e r e n c e between the stand volume before and a f t e r t h i n n i n g . Diameter growth i s computed as a f u n c t i o n of p r e v i o u s dbh, b a s a l area, and s i t e f o r a f i x e d p r e d i c t i o n p e r i o d . Moncatastrophic m o r t a l i t y i s computed as a f u n c t i o n of stand dbh and basal area per acre. Meyers' o r i g i n a l computer programs, c a l i b r a t e d f o r ponderosa pine {Pinus ponderosa Laws.) were only s l i g h t l y m o d i fied f o r the o p t i m i z a t i o n environment. The stand model was combined with an economic model based on the cost and b e n e f i t assumptions of Meyers (1973) (Table 1 ) 8 . A minimum commercial cut f o r saw l o g s was 1500 FBM/acre. Minimum commercial cut of roundwood was 300 c u b i c f e e t per acre from roundwood s a l e s and 100 c u b i c f e e t as a byproduct from saw l o g o p e r a t i o n s . The economic model r e t u r n s f o u r o b j e c t i v e f u n c t i o n values as measures of the performance of a management 8 The stand models and economic data s e t s i n t h i s t h e s i s were s e l e c t e d t o i l l u s t r a t e technigue, and are presented without comment on t h e i r v a l i d i t y . 81 sequence. - maximize volume production (CCF) - maximize volume production (MBF) - maximize discounted net v a l u e , one r o t a t i o n ($) - maximize discounted net value, i n f i n i t e s e r i e s of r o t a t i o n s ($) Table 1, Cost and b e n e f i t assumptions used i n the o p t i m i z a t i o n a n a l y s i s of Meyers' model. Costs Preccmmercial t h i n n i n g Cleanup (no salvage) Saw Log Sale Bound wood s a l e Seeding annual c o s t P r i c e s Bound wood stumpage Saw l o g Saw logs from t h i n n i n g 25.00 25.00 1.56 0.05 30.00 0. 20 2.50 15.00 12.75 $/acre $/acre $/MBF $/CCF $/acre $/year /CCF /MBF /MBF Due to the nature of Meyers* model, the only d e c i s i o n v a r i a b l e s t h at take on continuous v a l u e s are the growing stock l e v e l s . The c u t t i n g ages must be i n t e g e r m u l t i p l e s c f the i n t e r v a l c f p r o j e c t i o n , ten years f o r the ponderosa pine model. An o p t i m i z a t i o n problem i n v o l v i n g the growing stock l e v e l s of a t h i n n i n g regime i s d e s c r i b e d i n Meyers' 1971 paper: A f o r e s t manager wishes to determine the i n t e n s i t y of t h i n n i n g that w i l l maximize volume 82 prod u c t i o n i n board f e e t i n stands of s i t e index 70, A l t e r n a t i v e s c a l l i n g f o r more than one precommercial t h i n n i n g are unacceptable. Minimum commercial volumes are 320 c u b i c f e e t t o a 4-inch top and 1500 board f e e t . The manager expects t h a t the r e g e n e r a t i o n cuts w i l l r e s u l t i n a new stand t h a t c o n t a i n s 950 t r e e s per acre by age 30, with an average diameter of 4.8 i n c h e s . Meyers generated 50 y i e l d t a b l e s of combinations of i n i t i a l and subseguent t h i n n i n g l e v e l s , which were then examined to see which of them met the problem requirements. I t was found t h a t combinations of low i n i t i a l and low subsequent qrowinq stock l e v e l s or of hiqh i n i t i a l and in t e r m e d i a t e subsequent l e v e l s , produce the g r e a t e s t volumes with one precommercial t h i n n i n g . The response s u r f a c e was bimodal with one maxima at (80,80), i n i t i a l and subsequent t h i n n i n g l e v e l s r e s p e c t i v e l y , t h a t y i e l d e d 28 MBF. The other maxima found was at (120,100) and produced 29.3 MBF. This problem i s s o l v e d below using the o p t i m i z a t i o n s u p e r v i s o r . Although i t i s somewhat u n i n t e r e s t i n g , with o n l y two f r e e v a r i a b l e s and no requirement of the economic model, the problem w i l l serve t o demonstrate the technique. The uppercase m a t e r i a l provided below i s the a c t u a l user s e s s i o n as performed at a computer t e r m i n a l . The mode cf p r e s e n t a t i o n w i l l be demonstration user s e s s i o n s i n t e r s p e r s e d with e x p l a n a t o r y t e x t . The s e s s i o n s t a r t s with the user i n i t i a t i n g the ex e c u t i o n of the program SIMOPT and a s s i g n i n g some f i l e s . The POLICY f i l e i s always assigned and c o n t a i n s a s e l e c t i o n o f i n i t i a l p o l i c i e s 8 3 f o r the various models i n the model l i b r a r y . The temporary f i l e -POLICY w i l l be used t o save an i n t e r i m or optimum p o l i c y , and -YIELDTABLES w i l l s t o r e the managed stand y i e l d t a b l e s which Meyers' model produces. The s u p e r v i s o r prompts f o r in p u t with a guest i o n mark (?) . # $RUN SIMOPT 3=P0LICY 7=-P0LICY 8=-YIEIDTABLES # EXECUTION BEGINS ? READ PP#1 3 POLICY PP#1 , 19 PARAMETERS , 11 VARIABLES ARE READ IN. ? DISPLAY PP#1 TEST CASE : MEYERS MODEL PARAMETER D E S C R I P T I O N # VALUE 6 2.00 MEYER'S MODEL 7 80.00 BASE GSL 8 320.00 MIN KERCH CF VOL 9 1500.00 MIN MERCH VOL BF 10 10.00 INTERVAL OF PROJECTION 11 70. 00 SITE 12 30.00 INITIAL AGE 13 950.00 TPA AT THIN 14 4. 80 STAND DEH 15 25.00 PRE COM THIN COST /ACRE 16 25.00 CLEAN UP COST -NO SALVAGE 17 1. 56 TIMEER SALE COST /MBF 18 0. 05 TIMBER SALE COST /CCF 19 30. 00 SEEDING 20 0. 20 ANNUAL COSTS 21 2. 50 STUMPAGE /CCF 22 15.00 STUMPAGE /MEF 23 0.02 DISCOUNT RATE 24 2.00 IOBJ = MAX:MBF VARIAELE STEP LOWER UPPER # VALUE STATUS SIZE BOUND BOUND D E S C R I P T I O N 1 2.00 0 0.0 0.0 0.0 CYCLE 84 2 80.00 1 10.00 50.0 140. 0 GSI AT FIRST THIN 3 80.00 1 10.00 50.0 140. 0 MIN GSL AFTER THIN 4 11. 00 0 0. 0 0.0 0. 0 AGE FIRST CUT 5 0.50 0 0.0 0.0 0. 0 %GSL AT ROTATION 6 2. 00 0 0.0 0.0 G. 0 NEW INTERVAL 7 13. 00 0 0.G 0.0 0. 0 #2 CUT SHELTERWOOD 8 0.0 0 0.0 0.0 0. 0 %GSL AT 2ND CUT 9 0.0 0 0.0 0.0 0.0 INTERVAL 10 0.0 0 0.0 0.0 0. 0 #3 CUT SHELT ERWOOD 11 2.00 0 0.0 0.0 0. 0 1ST THIN INTERVAL The READ command reads the p o l i c y with the l a b e l PP#1 from the p o l i c y f i l e a ssigned to l o g i c a l u n i t 3. The p o l i c y i s then DISPLAYed. The parameter l i s t d e f i n e s those elements o f the s i m u l a t i o n model which would not l i k e l y be d e c i s i o n v a r i a b l e s , An i n t e g e r number i s assigned to each parameter and i s used to i d e n t i f y the parameter i n s u p e r v i s o r commands. For example, parameter 24 i s the choice of performance c r i t e r i o n . The value 2 i n d i c a t e s t h a t the o b j e c t i v e i s to maximize volume production i n thousands of board f e e t . The v a r i a b l e l i s t provides an analogous i n t e g e r i d e n t i f i e r and i n i t i a l values of the v a r i a b l e s , as w e l l as i n f o r m a t i o n f o r the s e g u e n t i a l simplex a l g o r i t h m . The •status* of a v a r i a b l e i s 1 i f i t i s f r e e to take on new values i n the o p t i m i z a t i o n process, or 0 i f i t i s f i x e d . The *step s i z e 1 d e f i n e s the dimensions of the o r i g i n a l simplex, and the * bounds' are the e x p l i c i t c o n s t r a i n t s d e s c r i b e d e a r l i e r . In the p o l i c y DISPLAYed above, only the growing stock l e v e l a f t e r the f i r s t t h i n n i n g , and the growing stock l e v e l s a f t e r subsequent t h i n n i n g s , are f r e e to vary i n the o p t i m i z a t i o n process. ? SIMULATE VOLUME OF CUT EETURN A C T I V I T Y AGE BP. CF. $ PEECCMMEECIAL THIN 30. -50.0 ROUND HOOD SALE 50. 0.0 362. 2 -16.1 SAB LOG SALE 70. 1650.0 751.0 24.9 SAW LOG SALE 90. 3466.8 865.3 38.8 SAW LOG SALE 110. 106 37.6 2114.1 124.6 SAW LOG SALE 130. 12187.8 2007.5 172.4 OBJECTIVE VALUE IS 27.94 The p o l i c y i s SIMULATEd above. A l l volumes and d o l l a r values are expressed on a per acre b a s i s . Meyers' model produces a managed stand y i e l d t a b l e on the s c r a t c h f i l e -YIELDTAELES, and the economic model produces a sh o r t r e p o r t at the t e r m i n a l . The y i e l d t a b l e i s i n c l u d e d i n Appendix V ( P o l i c y #1). From the r e p o r t s one sees that the t h i n n i n g at age 30 i s precommercial. The second t h i n n i n g a t age 50 produces enough merchantable cub i c f o o t volume f o r a roundwood s a l e (>320 c u b i c feet) but no sawtimber. The subsequent cuts y i e l d merchantable volumes of saw timber. Note that the round wood s a l e was u n p r o f i t a b l e ($ -16.10) as h a r v e s t i n g c o s t s exceeded b e n e f i t s . The s a l e of the roundwood helped defray the c o s t of the t h i n n i n g c u t . ? OPTIMIZE 30 V A R I A B L E S # Z* 2 3 0 27.94 2 80.00 80. 00 1 29.395 80. 00 90. 00 2 30.000 70. 00 90. 00 3 30.000 70.00 90.00 4 30.040 75.00 92. 50 5 30.040 75.00 92. 50 6 30.107 73.75 91.88 7 30.195 74. 38 92. 19 8 30.195 74.38 92. 19 9 30.310 73. 91 92. 58 10 30.356 74. 53 92. 89 11 30.356 74.53 92, 89 12 30.356 74.53 92.89 13 30.356 74. 53 92. 89 14 30.356 74.53 92. 89 15 30.356 74. 53 92. 89 AFTER 15 ITERATIONS,THE BEST RETURN IS 30.356 FREE VARIABLES : 74.53 92.89 ? SIMULATE A C T I V I T Y AGE VOLUME OF CUT RETURN BF. CF. $ PRECOMMERCIAL THIN 30. -50.0 PREC CM MEBCIAL THIN 50. -50.0 SAW LOG SALE 70. 1568.0 794.5 25.4 SAW LOG SALE 90. 3547.9 924.4 39.7 SAW LOG SALE 110. 11288.1 2313.8 132.8 SAW LOG SALE 130. 13952.0 2355.4 199.3 OBJECTIVE VALUE IS 30. 36 The OPTIMIZE procedure was invoked f o r a maximum of 30 i t e r a t i o n s but convergence had occurred a f t e r 15 i t e r a t i o n s . The o b j e c t i v e f u n c t i o n of maximizing volume y i e l d s i n c r e a s e d from 27.942 to 30.356 MBF as the p o l i c y changed from an i n i t i a l and subseguent GSL of 80 and 80 r e s p e c t i v e l y , t o growing stock l e v e l s of 74.5 and 92.9. S i m u l a t i o n cf t h i s optimal p o l i c y demonstrates t h a t i t i n v o l v e s two precommercial t h i n n i n g s , v i o l a t i n g the problem s p e c i f i c a t i o n s . The SET command i s used below t o i n c r e a s e the i n t e r v a l a f t e r the f i r s t t h i n n i n g ( v a r i a b l e 11) from 2 t o 4 decades, e l i m i n a t i n g the t h i n n i n g at 70 years. 87 ? SET V 11 4. ? OPTIMIZE 30 v a R i A B L E # Z* 2 3 0 30.011 74. 53 92. 89 1 30.011 74.53 92. 89 2 30.075 79.53 95.39 3 30.950 75.7 8 98. 52 4 30.950 75.76 98.52 5 30.950 75.78 98. 52 6 30.950 75.78 98.52 7 30.950 75.78 98. 52 8 3C.950 75.78 S8. 52 9 31.139 75.70 99. 41 10 31. 139 75.70 99.41 11 31.226 76. 28 100.29 12 31.226 76.28 100.29 13 31.237 75. 94 99. 61 14 31,237 76. 10 99.64 15 31. 237 76. 10 9 9. 64 16 31.237 76. 10 99.64 17 31.237 76. 10 99. 64 18 31.237 76.10 99.64 19 31. 237 76. 01 99, 54 AFTER 19 ITERATIONS,THE BEST RETURN IS 31.237 FREE VARIAELES : 76.10 99.64 ? SIMULATE A C T I V I T Y AGE PRECCMMERCIAL THIN 30. SAW LOG SALE 70. SAW LOG SALE 90. SAW LOG SALE 110. SAW LOG SALE 130. VOLUME OF CUT BF. CF. 1621.0 3263.7 1 1724.1 14627.8 923.9 917.0 2532.7 2539.2 RETURN $ -50.0 28.1 36.5 138.5 209.6 OBJECTIVE VALUE IS 31.24 ? WRITE 7 ? STOP # EXECUTION TERMINATED Furthe r o p t i m i z a t i o n of the p o l i c y without the t h i n n i n g a t age 70 r e s u l t s i n an adjustment of the i n i t i a l and subseguent 88 growing stock l e v e l s to 76 and 99.5 sg. f t . , r e s p e c t i v e l y , and an improvement i n y i e l d to 31.237 MBF. Si m u l a t i o n of the p o l i c y shows only one precommercial t h i n n i n g with subsequent c u t s as p r o f i t a b l e saw l o g s a l e s . In comparison, Meyers' a n a l y s i s r e s u l t e d i n a best p o l i c y t h a t produced 29.3 MBF a t growing stock l e v e l s of 120 and 100 sg. f t . The y i e l d t a b l e i s i n c l u d e d i n Appendix V ( P o l i c y #2). The optimal p o l i c y i s then saved on the temporary f i l e with the WHITE command and e x e c u t i o n of s u p e r v i s o r i s terminated with the STOP command. The problem was f u r t h e r analysed a f t e r changing the o b j e c t i v e f u n c t i o n from maximizing board f o o t y i e l d s to maximizing the net present worth. The p o l i c y was re l o a d e d and opti m i z e d from (80,80), with the i n t e r v a l a f t e r the f i r s t -chinning s et at 4 decades. The o p t i m i z a t i o n a l g o r i t h m converged at a subseguent and i n i t i a l GSL o f (115,108), with a present net worth of $27.13 per a c r e . Knowledge of the op t i m a l p o l i c y i s u s u a l l y i n s u f f i c i e n t i n f o r m a t i o n f o r the d e c i s i o n maker; one a l s o needs t o know the behavior of the o b j e c t i v e f u n c t i o n at p o i n t s nearby, A rough s e n s i t i v i t y a n a l y s i s of the o b j e c t i v e f u n c t i o n with r e s p e c t to the d e c i s i o n v a r i a b l e s can be accomplished with the SET command. The subsequent t h i n n i n g GSL was Fi x e d at 108 s q . f t . and the i n i t i a l GSL was v a r i e d over the range 108 - 117. Sim u l a t i o n runs were performed at 1 s q . f t . i n t e r v a l s and the o b j e c t i v e f u n c t i o n values are represented i n F i g u r e 8 with a s o l i d l i n e . There are two f e a t u r e s of i n t e r e s t i n Figure 8. F i r s t , the o p t i m i z a t i o n a l g o r i t h m has converged t o a suboptimum 89 1 1 1 1 1 i i 1 1 1 1 1 1 1 1 1 r 105 107 109 111 113 115 117 119 Intensity of f i r st thin ( G r o w i n g stock l eve l after thinning) Figure 8. S e n s i t i v i t y of Meyers' model to ranging of the i n t e n s i t y of the f i r s t thinning. 90 po i n t (115,108)- while at l e a s t one b e t t e r p o i n t e x i s t s (111,108). Second, the s u r f a c e of the o b j e c t i v e f u n c t i o n a p p a r e n t l y has two sharp d i s c o n t i n u i t i e s , at (112,108) and (116,108). One of these 'deep r a v i n e * d i s c o n t i n u i t i e s s e p arates the l o c a l optima, found by the o p t i m i z a t i o n a l g o r i t h m , from the b e t t e r p o i n t a t (111,108) and may account f o r the f a l s e convergence. The nature of the d i s c o n t i n u i t y can be d i s c e r n e d from the s i m u l a t i o n t a b l e s . At the apparent optimum (111,108), the t h i n n i n g at age 70 y i e l d s 1742.7 b d . f t . o f sawtimber: ? SIKULATI A C T I V I T Y AGE VOLUME OF CUT SETUBN BF. CF. $ P B EC CM M EEC I AL THIN 30. -50.0 SAW LOG SALE 70. 1742.7 1211.8 34.1 SAW LOG SALE 90. 4481.4 1019.1 50.1 SAW LOG SALE 110. 11561.8 2688.8 136.8 SAW LOG SALE 130. 14473.7 2639.4 206.7 OBJECTIVE VALUE IS 28.43 This i s s u f f i c i e n t volume f o r a sawlog s a l e and a r e t u r n of $34.00 per acre i s produced. However, by i n c r e a s i n g the GSL a t the f i r s t t h i n by 1 to 112 s q . f t . , the y i e l d of the second t h i n n i n g i s reduced to 1493 b d . f t . , which i s s l i g h t l y l e s s than the minimum l i m i t f o r a sawlog s a l e (1500 b d . f t . ) . Consequently, a l e s s p r o f i t a b l e roundwood s a l e was simulated, which r e t u r n e d o n l y $3.40 per ac r e : 91 ? SIMULATE A C T I V I T Y PRECCMMERCIAL THIN ROUND WOOD SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE OBJECTIVE VALUE IS VOLUME AGE BF. 30. 70. 1493.6 90. 4262.6 110. 11531.0 130. 14393.3 11.09 OF CUT RETURN CF. $ -50.0 1157.9 3.4 1028.1 47.7 2690.6 136.2 2644.9 205.3 This d i s c o n t i n u i t y was removed from the o b j e c t i v e f u n c t i o n by SETting the minimum board foot volume c o n s t r a i n t to ze r o , and the o p t i m i z a t i o n a l g o r i t h m was r e s t a r t e d . ? OPTIMIZE 30 # Z* 0 27.579 1 27.579 2 27.579 3 27.57 9 4 27.579 5 27.579 6 27.579 7 28. 248 8 2 9.089 9 29.089 10 29.089 11 29.089 12 29.089 13 29.089 14 29.089 15 29.089 16 29.089 V A R I A B L E S 2 3 115.00 108.00 115.00 108.00 115.00 108.00 115.00 108. 00 115.00 108.00 115. 00 108. 00 115.00 108.00 110. 00 108.00 11 1.09 108. 31 111.09 108.31 111.09 108.31 11 1. 09 108. 31 11 1.09 108. 31 111.21 108.48 111.21 108.48 111.10 108.44 110. 90 108. 35 AFTER 16 ITERATIONS,THE BEST RETURN IS 29.089 FREE VARIAELES : 111.10 108.44 ? SIMULATE 92 A C T I V I T Y AGE VOLUME OF CUT BP. CF. RETURN $ SAW LOG SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE PRECOMMERCIAL THIN 30. 70. 90. 110. 130. 1674.5 4591.2 11666.1 14621.6 1188.9 1036.6 2701.8 2675.7 -50.0 33.4 51.4 137.9 208.9 OBJECTIVE VALUE IS 29.09 With the d i s c o n t i n u i t i e s removed, the a l g o r i t h m converged at the apparent optimum (Appendix V, P o l i c y #4). The s e n s i t i v i t y a n a l y s i s was repeated over the o r i g i n a l range and the smooth s u r f a c e i s represented i n Figure 8 with the s o l i d l i n e . The above e x e r c i s e demonstrates a number of ways t h i s approach might f a i l . Multimodal o b j e c t i v e f u n c t i o n s are not d i f f i c u l t to c o n c e i v e of where a stand management seguence i s being o p t i m i z e d . For example, i f the d e c i s i o n v a r i a b l e s are the i n t e n s i t y c f the f i r s t and second t h i n n i n g , then one can imagine a l o c a l optima where a heavy i n i t i a l t h i n n i n g y i e l d s low g u a l i t y wood, g r e a t l y r e d u c i n g the y i e l d a v a i l a b l e f o r the second t h i n n i n g , but i s d e s i r a b l e because the d i s c c u n t r a t e favours e a r l y t h i n n i n g r e t u r n s . A second optimum might occur where a moderate f i r s t t h i n n i n g r e s u l t s i n high q u a l i t y and high value wood i n the second t h i n n i n g . The multimodal s i t u a t i o n d e s c r i b e d i n the Meyers t e s t problem r e s u l t e d from a d i s c o n t i n u i t y caused by a c o n s t r a i n t i n the economic model. Models t h a t are to be analysed by t h i s 9 3 approach must be examined c a r e f u l l y f o r c o n s t r a i n t s , l o g i c a l v a r i a b l e s , or s t a t e t r a n s i t i o n f u n c t i o n s that w i l l r e s u l t i n d i s c o n t i n u i t i e s i n the o b j e c t i v e f u n c t i o n , Models that g i v e a c c e p t a b l e r e s u l t s as simple s i m u l a t i o n s may not be we l l enough behaved f o r o p t i m i z a t i o n . One might guestion the v a l i d i t y of a stand model that p r e d i c t s a r e d u c t i o n of 250 b d . f t . at a t h i n n i n g of a 70 year o l d stand, as a r e s u l t of a 1 s q . f t . change i n the t h i n n i n g i n t e n s i t y of the stand at age 30. Of course, Meyers 1 model was not designed to be used i n t h i s manner, but the above a n a l y s i s serves t o demonstrate some of the p i t f a l l s c f the approach. 4.1.4 Test Case: Goulding's Model Goulding's D o u g l a s - f i r growth model was c h a r a c t e r i z e d i n Chapter 2 as an e m p i r i c a l , s i n g l e t r e e , d i s t a n c e independent model. A b r i e f d e s c r i p t i o n of some of the model's processes i s necessary to e v a l u a t e i t s u t i l i t y i n c o n j u n c t i o n with the o p t i m i z a t i o n s u p e r v i s o r . The s t a t e v a r i a b l e s of the growth model are the stand age and a dbh vector of a r e p r e s e n t a t i v e sample of the stand ( u s u a l l y a 1/4 acre plot) . The s i m u l a t i o n i s i n i t i a t e d at age 20 by generating an average stand dbh and then a s s i g n i n g dbh to the elements of the dbh s t a t e v e c t o r a c c o r d i n g to an e m p i r i c a l d i s t r i b u t i o n . Each c y c l e of growth begins by computing the dominant height of the stand as a f u n c t i o n of age and s i t e 94 index, The expected m o r t a l i t y (stems per acre) i s computed from an e m p i r i c a l r e g r e s s i o n f u n c t i o n of stand c h a r a c t e r i s t i c s , i n c l u d i n g age, s i t e index, b a s a l area and stems per acre. Goulding adds a s t o c h a s t i c component formed by sampling a random d e v i a t e from an e m p i r i c a l d i s t r i b u t i o n d e f i n e d by the standard e r r o r of the m o r t a l i t y r e g r e s s i o n r e l a t i o n s h i p . T h i s t o t a l m o r t a l i t y i s then d i s t r i b u t e d between the stems represented by the dbh s t a t e v e c t o r . The a l l o c a t i o n of m o r t a l i t y among the elements of the s t a t e vector i s a l s o a s t o c h a s t i c process. The average dbh f o r the stand i s a l s o computed from a r e g r e s s i o n f u n c t i o n of stand c h a r a c t e r i s t i c s ; s i t e index, dominant height, average dbh, basal area, and stems per acre. The average stand dbh increment i s then a l l o c a t e d between the elements of the s t a t e v e c t o r i n a continuous manner, ac c o r d i n g to an e m p i r i c a l r e l a t i o n s h i p . The dbh increment and m o r t a l i t y c a l c u l a t i o n s are repeated f o r each growth p e r i o d of l e s s than s i x years. Goulding p r o v i d e s f o r two modes of t h i n n i n g , both i n v o l v i n g o p e r a t i o n s on the dbh ve c t o r . The f i r s t , termed a low t h i n n i n g , removes t r e e s i n ascending order from a s p e c i f i e d minimum dbh u n t i l a d e s i r e d amount of b a s a l area has been removed. The second, termed a crown t h i n n i n g , removes t r e e s from a pre d e f i n e d dbh c l a s s . The major m o d i f i c a t i o n made to Goulding*s model f o r o p t i m i z a t i o n a n a l y s i s was to remove the random r e s i d u a l m o r t a l i t y component. Goulding determined that the i n t e r a c t i o n of t h i s s t o c h a s t i c component of m o r t a l i t y with the other elements 95 of the model r e s u l t e d i n a s m a l l decrease i n y i e l d . However, he allowed that 'the d i f f e r e n c e between a s t o c h a s t i c model such as t h i s one, and a d e t e r m i n i s t i c one would be n e g l i g i b l e when used to o b t a i n average v a l u e s ' 9 . The r a t i o n a l e f o r the o r i g i n a l i n c l u s i o n of the s t o c h a s t i c element i s u n c l e a r , but the a d d i t i o n a l e f f o r t needed f o r repeated e v a l u a t i o n s c f the s t o c h a s t i c model i s c l e a r l y u n d e s i r e a b l e i n an o p t i m i z a t i o n framework. Because the second component of v a r i a b i l i t y , the p r o b a b i l i s t i c a l l o c a t i o n of m o r t a l i t y among the stems of the dbh v e c t o r , was an i n t e g r a l part of the model t h a t could not be removed, e l i m i n a t i o n of the random r e s i d u a l m o r t a l i t y does not r e s u l t i n a wholly d e t e r m i n i s t i c model. Goulding's o r i g i n a l programs were g e n e r a l l y r e o r g a n i z e d i n t o s u b r o u t i n e s with e n t r y p o i n t s to c r e a t e twenty year o l d stands, grow stands to any age, perform t h i n n i n g or c l e a r c u t s , and compute volumes to any u t i l i z a t i o n standards. The stand growth model was combined with a simple economic model based cn Hoyer's (1975) cost and b e n e f i t assumptions (Table 2) . The c u b i c f o o t volumes produced by Goulding's model were converted to board f e e t by assuming an average lumber r e c c v e r y f a c t o r of 7. The minimum merchantable volume t h a t was f e a s i b l e to t h i n n i n g was c o n s t r a i n e d to be 8 c u n i t s per acre. » Goulding (1972, p. 115) 96 Table 2, Cost and b e n e f i t assumptions used with of Goulding's model (Hcyer (1975) ) . The economic model r e t u r n s s i x o b j e c t i v e f u n c t i o n values as measures of performance of a management seguence: mean annual increment of c l o s e u t i l i z a t i o n volume, i n c u n i t s mean annual increment of c l o s e u t i l i z a t i o n volume, i n MBF mean annual increment of i n t e r m e d i a t e u t i l i z a t i o n volume, i n c u n i t s mean annual increment of in t e r m e d i a t e u t i l i z a t i o n volume, i n MBF present net worth of c l o s e u t i l i z a t i o n volume present net worth of in t e r m e d i a t e u t i l i z a t i o n volume The f o l l o w i n g user s e s s i o n e x p l o r e s the op t i m a l neighborhood p o l i c y space of a m u l t i p l e commercial t h i n n i n g management sequence. The example w i l l a l s o demonstrate some of the c h a r a c t e r i s t i c s o f Goulding's model i n the framework of o p t i m i z a t i o n a n a l y s i s . The s u p e r v i s o r program i s i n i t i a t e d i n Ease year annual c o s t /acre stand e s t a b . c o s t /acre precommercial t h i n cost /acre c l e a r c u t s a l e cost /MBF t h i n n i n g s a l e cost /MBF p r i c e /MBF $ 1971 $ .92 $ 56.06 $ 54.5 8 $ 3.89 $ 8.75 14.05 + 1.34 dbh 97 e x a c t l y the same way as with the Meyers' t e s t case, except t h a t a s c r a t c h f i l e i s assigned to l o g i c a l u n i t 9 t o hold the stand t a b l e s generated by Goulding's model. # $RUN SIMOPT 3=F0LICY 9=-STANETABIES # EXECUTION BEGINS ? READ DF#1 3 POLICY DF#1 , 18 PARAMETERS , 7 VARIABLES ARE READ IN. ? DISPLAY DF#1 TEST CASE #1 : GOULDING'S MODEL * « • • • • • • • • • • • • • • • • • * • • • PARAMETER D E S C R I P T I O N # VALUE 6 1, 00 MODEL TYPE 7 150. 00 SITE INDEX 8 0. 25 PLOT SIZE 9 7. 10 DBH CU 10 11. 10 DBH IU 1 1 0.06 DISCCUNT RATE 12 56.00 ESTABLISHMENT COST 13 54.00 STAND STOCKING CONTROL-14 0.92 ANNUAL COST 15 3. 79 CLIARCUT COST /MBF 16 8.75 THIN COST /MBF 17 8.00 MIN MERCHANTABLE CCF/ACRE 18 0.10 DBH < 0.75 19 0. 25 0.75 <= DBH < 1.00 20 0.32 1.00 <= DBH < 1.2 5 21 0. 25 1.25 <= DBH < 1.50 22 0.07 1.50 <= DBH 23 1.00 OBJECTIVE : MAX MAI CU VARIABLE # VALUE STATUS 1 100.00 1 2 40.00 1 3 7.00 0 4 90. 00 0 5 20.00 1 6 io.oo 0 7 90. 00 1 STEP LOSER UPPER SIZE BOUND BOUND 25.00 50.0 400.0 3.00 21.0 100. 0 2.00 1.0 4. 0 0.0 0.0 0. 0 5.00 15.0 30.0 2.00 3.0 15. 0 5.00 30.0 120. 0 D E S C R I P T I O N STOCKING AT AGE 20 AGE INITIAL THIN MAX # OF THINNINGS MAX AGE OF THIN INTENSITY THIN %EA CUTTING INTERVAL AGE HARVEST CUT 98 The p o l i c y BEAD i n and DISPLAYed above d e f i n e s a high s i t e D o u g l a s - f i r stand. Present net worth i s to be c a l c u l a t e d with a d i s c o u n t r a t e of 6%. The c u r r e n t index of performance i s the mean annual increment of c l o s e u t i l i z a t i o n volume. The i n t e n s i t y of each t h i n n i n g ( v a r i a b l e 5) i n percent of stand b a s a l area i s d i s t r i b u t e d among f i v e dbh c l a s s e s according to the p r o p o r t i o n s given i n parameters 18 - 22. The t h i n n i n g i n t e n s i t y i s a f r e e v a r i a b l e i n the o p t i m i z a t i o n a n a l y s i s but i s c o n s t r a i n e d between 15% and 30% of the stand b a s a l area. The other t h r e e f r e e v a r i a b l e s are the d e n s i t y at age 20 (100 t r e e s per 1/4 acre p l o t , c o n s t r a i n e d t o the i n t e r v a l 50 - 400 t r e e s ) , the age of the i n i t i a l t h i n n i n g (40 years, c o n s t r a i n e d to 21 - 100 y e a r s ) , and the age of the harvest c u t (90 years, c c n s t r a i n e d to 30 120 y e a r s ) . The time i n t e r v a l between cuts i s f i x e d at 10 ye a r s . The SIMULATE command r e t u r n s the y i e l d t a b l e to the t e r m i n a l screen and the stand t a b l e to the s c r a t c h f i l e -STANDTABIES. The stand t a b l e i s i n Appendix VI ( P o l i c y #1). ? SIMULATE MAIN CBOP SI 150 * GBOSS * THINNINGS * * * * i | n t M * * * * * * i * * # * * * * * * * * * * * * * + ***** ** ********** ************* 20 400 4.7 53. 901. * 901. 45 * 40 216 10.2 136. 4634. * 5895. 147 * 92 8.3 39. 1261. 50 143 12.8 141. 54 97. * 8146. 1 63 * 57 10. 4 37. 1388. 60 97 15.2 136. 5809. * 10051. 168 * 41 12.4 38. 1593. 70 6 4 18.0 124. 56S6. * 11569. 165 * 28 14. 6 36. 1631. 90 6 3 20.7 161. 8182. * 14055. 156 * OBJECTIVE VALUE IS 1.21 ? SIMULATE 5 OBJECTIVE VALUE IS 1.30 99 The stand and y i e l d t a b l e s are the r e s u l t of a s i n g l e s i m u l a t i o n and the o b j e c t i v e value was c a l c u l a t e d to be 1.21 CCF/year. Sfhen the p o l i c y i s SIMULATEd with 5 r e p e t i t i o n s using d i f f e r e n t random number s t r i n g s , no t a b l e s are produced and the average MAI i s found to be 1,3 CCF/year. Note t h a t the t h i n n i n g s a t ages 40 and 50 years v i o l a t e the c o n s t r a i n t t h a t the minimum IU volume f e a s i b l e t o t h i n n i n g i s 8 CCF (parameter 17). The index of performance was reSET t o be the net present worth of CO volume. The p o l i c y was then OPTIMIZEd f o r 40 i t e r a t i o n s . The l a s t 20 i t e r a t i o n s were performed with 3 r e p e t i t i o n s per i t e r a t i o n . ? SET P 23 5. ? OPTIMIZE 20 V A R I A B L E S # Z* 0 -37.836 1 -20.798 2 -20.017 3 -19.439 4 -9.085 5 -9.085 6 -9.085 7 -9.085 8 -8.140 9 -8.140 10 -8.140 11 -8.140 12 -8.140 13 -7.784 14 -7.784 15 -7.784 16 -7.245 17 -7.245 18 -7.145 19 -5.456 20 -5.456 100.00 40,00 125.00 40, 00 118.75 42.25 121. 88 4 2. 63 149.22 37.66 149.22 37. 66 149.22 37,66 149.22 37,66 190.67 37.52 190. 67 37. 52 190. 67 37. 52 190. 67 37. 52 190.67 37. 52 164. 1 1 37. 54 164.11 37.54 164. 11 37. 54 169.80 36,99 169. 80 36. 99 169.72 36. 80 180. 19 37. 16 180.19 37.16 1 2 20, 00 20. 00 23. 75 24. 38 29. 84 2 9. 84 29. 84 29. 84 30. 00 30. 00 30. 00 30. 00 30. 00 28. 68 28.68 28. 68 29. 42 29. 42 29.79 29. 90 29. 90 5 90. 00 90. 00 80.00 85. 00 78.75 78. 75 78.75 78. 75 65.70 65.70 65. 70 65. 70 65.70 70. 30 70.30 70. 30 71.42 71. 42 70. 81 68. 25 6 8. 25 7 AFTER 20 ITERATIONS,THE BEST RETURN IS -5.456 FREE VARIAELES : 180.19 37.16 29.90 68.25 ? OPTIMIZE 20 3 100 V A B I A B L E S # Z* 1 2 5 7 0 -14.714 180.19 37. 16 29. 90 68.25 1 -13.189 180. 19 40. 16 29. 90 68. 25 2 -11. 392 130. 19 39.41 29. 97 72. 00 3 -9.884 155. 19 39. 79 29. 99 65. 13 4 -9.884 15 5.19 39. 79 29. 99 65.13 5 -7.164 170.82 38. 15 29. 97 68. 33 6 -7.164 170. 82 38. 15 29. 97 6 8. 33 7 -7.164 170.82 38. 15 29. 97 6 8.33 8 -7. 164 170.82 38. 15 29. 97 68. 33 9 -6.649 153.75 37. 13 29. 99 68. 48 10 -6.649 153.75 37. 13 29. 99 68.48 11 -6.003 156.07 37. 59 29. 99 68. 04 12 -6.003 156.07 37. 59 29. 99 68. 04 13 -6.003 156.07 37. 59 29. 99 68. 04 14 -5.895 155, 25 37.98 29, 99 68.46 15 -5.895 155.25 37. 98 29. 99 68. 46 16 -5. 895 155.25 37. 98 2 9. 99 68.46 17 -5.795 155. 66 37. 78 29. 99 68. 25 18 -5.78 8 155.59 37. 76 29.99 68. 26 19 -5.765 155.14 38.01 29. 99 68. 17 20 -5.765 155.14 38.01 29. 99 68. 17 AFTEE 20 ITEBATIONS,THF BEST RETURN IS -5.765 FREE VARIAEIES : 155.14 38.01 29.99 68.17 ? SIMULATE MAIN CBOP SI 150 AFTEE THINNING AV. TOTAL AGE STEMS DEH EA VOL * GBOSS * THINNINGS * PRODUCTION * * TOTAL * AV. TOTAL * VOL MAI * STEMS DBH BA VOL ************************************************************* 20 620 4.3 68. 1109. * 38 272 9.1 136. 4437. * 48 144 12.0 125. 4741. * 58 80 15.1 109. 4605. * 68 80 16.7 134. 6070. * 1109. 55 * 6266. 165 * 8698. 181 * 10569. 1 82 * 12033. 177 * 192 108 7.2 9.5 59 11.7 60. 1830. 58. 2127. 49. 2006. OBJECTIVE VALUE IS -5.96 The index of performance was reSET to be the net present worth of CU volume. The p o l i c y was then OPTIMIZEd f o r 40 i t e r a t i o n s . The l a s t 20 i t e r a t i o n s were performed with 3 r e p e t i t i o n s per i t e r a t i o n . The i n i t i a l p o l i c y had a very low net present worth of 101 $ -37.836, probably due to the u n p r o f i t a b l e t h i n n i n g s at 40 and 60 years. (If a t h i n n i n g r e s u l t s i n a volume below the minimum, the economic model charges the t h i n n i n g c o s t but does not c a l c u l a t e the r e t u r n on s a l e s . ) the most dramatic changes t o the p o l i c y were made i n the f i r s t 4 i t e r a t i o n s . By i n c r e a s i n g the stand d e n s i t y at age 20 from 100 to 149 t r e e s per 1/4 a c r e , and i n c r e a s i n g the i n t e n s i t y of the t h i n n i n g s to the upper l i m i t (30% of stand b a s a l a r e a ) , the p r o f i t a b i l i t y of the e a r l y t h i n n i n g s was improved. A f t e r 40 i t e r a t i o n s , the p o l i c y appeared to have s t a b l i z e d with 155 t r e e s per 1/4 acre and a s l i g h t r e d u c t i o n i n the age of f i r s t t h i n n i n g t o 38 years. The maximum t h i n n i n g i n t e n s i t y (30% stand b a s a l area) was p r e s c r i b e d , and the age of harvest cut reduced s u b s t a n t i a l l y to 68 years, e l i m i n a t i n g the t h i n n i n g at age 70. The stand t a b l e f o r t h i s p o l i c y (Appendix VI, P o l i c y #2) r e v e a l s that the f i r s t t h i n n i n g i s s t i l l below the 10 volume l i m i t . Although the o p t i m i z a t i o n a l g o r i t h m appears to have s t a b i l i z e d , i t i s reasonable to suspect t h a t e l i m i n a t i n g the t h i n n i n g at 38 years would improve the net present worth. T h i s s i t u a t i o n demonstrates the n e c e s s i t y of human i n t e r v e n t i o n i n t h i s type of a n a l y s i s . The a l g o r i t h m has converged t o a l o c a l maximum, with no way of s y s t e m a t i c a l l y e l i m i n a t i n g the t h i n n i n g at 38 years without encountering an in t e r m e d i a t e p o l i c y with a lower net present worth. E x p l o r a t i o n of the p o l i c y region with the f i r s t t h i n s l i g h t l y above 38 years, would i n v o l v e p o l i c i e s where the f i r s t t h i n n i n g was s t i l l u n p r o f i t a b l e and the next ( p r o f i t a b l e ) t h i n n i n g at 48 years would have to be delayed due to the constant c u t t i n g i n t e r v a l of 10 years. The j o i n t e f f e c t 102 of t h i s delay and the discount r a t e would be to reduce the NPw even f u r t h e r . Consequently, there i s no pathway on the NPH s u r f a c e from the c p t i m i a l two t h i n n i n g p o l i c y to a o n e - t h i n n i n g p o l i c y that i s continuous and nondecreasing, fin i n t i m a t e knowledge of the stand model, the economic model, and the o p t i m i z a t i o n a l g o r i t h m i s necessary to recognize and i n t e r p r e t these l o c a l optima. ? SET V 2 48. ? SIMULATE MAIN CHOP SI 150 * GROSS * THINNINGS AETEB THINNING * PRODUCTION * AV. TOTAL * TOTAL * A?. TOTAL AGE STEMS DBH EA VOL * VOL MAI * STEMS DBH BA VOL **************************************************** 20 620 4.3 68. 1109. * 1109. 55 * 48 208 11.3 159. 5965. * 8533. 178 * 152 8.8 71. 2568. 58 117 13.9 137. 5704. * 10762. 186 * 83 11.2 62. 2490. 68 116 15.4 166. 7473. * 12531. 184 * OBJECTIVE VALUE IS 7.09 The SET command was used to ad j u s t the age of the f i r s t t h i n n i n g at 48 years. S i m u l a t i o n of the p o l i c y shows t h a t the volume of the f i r s t t h i n n i n g i s above the minimum l i m i t , and th a t the net present worth i s g r e a t l y i n c r e a s e d . ? OPTIMIZE 20 3 V A R I A B L E S # Z* 1 2 5 7 0 5. 496 155. 14 48, 00 29. 99 68. 17 1 5.496 155. 14 48.00 29. 99 68. 17 2 496 155. 14 48. 00 29. 99 68. 17 3 5. 653 155.14 48.00 29. 99 7 0.67 4 7. 114 148.89 46. 88 29. 99 70. 05 5 7. 114 148.89 46. 88 29. 99 7 0.05 6 7. 153 152. 01 47. 44 29. 99 69. 11 7 7. 230 152.60 47. 17 29. 99 69.64 8 7. 230 152.60 47. 17 29. 99 69. 64 9 7. 230 152. 60 47. 17 29. 99 69.64 10 7. 230 152. 60 47. 17 29. 99 69. 64 11 7. 230 152.60 47. 17 29. 99 69.64 12 7. 452 152. 57 47. 23 29. 99 68. 94 13 •7. 452 152.57 47.23 29. 99 68. 94 14 7. 452 152.57 47, 23 29. 99 68. 94 15 7. 452 152.57 47. 23 29. 99 68. 94 16 7. 452 152.57 47. 23 29. 99 6 8. 94 17 7. 452 152.57 47.23 29.99 68.94 18 7. 749 152.70 47. 22 29. 99 68. 58 19 7. 973 152.56 47. 19 29. 99 68.56 20 9. 578 152. 70 47. 35 29. 99 67. 48 AFTER 20 ITERATIONS,THE BEST RETURN IS 9.578 FREE VAEIAELES : 152. 70 47. 35 29. 99 67.48 OPTIMIZE 20 3 V A R I A B L E S # Z* 1 2 5 7 0 9. 578 152.70 47.35 29. 99 67.48 1 9. 578 152.70 47. 35 29. 99 67. 48 2 9. 578 152.70 47. 35 29. 99 67. 48 3 9. 578 152. 70 47. 35 29. 99 67. 48 4 9. 578 152.70 47. 35 29.99 67.48 5 9. 578 152.70 47. 35 29. 99 67. 48 6 9. 578 152,70 47. 35 29. 99 67.48 7 9. 578 152.70 47. 35 2-9. 99 67. 48 8 9. 578 152.70 47.35 29. 99 67. 48 9 9. 578 152. 70 47. 35 29. 99 67. 48 10 9. 578 152. 70 47. 35 29.99 67.48 11 9. 578 152. 70 47. 35 29. 99 67. 48 12 9. 804 152.11 47. 27 29. 99 67.34 13 9. 804 152. 11 47. 27 29. 99 67. 34 14 9. 804 15 2. 11 47. 27 29. 99 67.34 15 9. 804 152. 11 47. 27 29. 99 67. 34 16 9. 804 152.11 47. 27 29. 99 67. 34 17 9. 804 152. 11 47. 27 29. 99 67. 34 18 9. 804 152. 11 47. 27 29. 99 67.34 19 9. 850 152.31 47. 25 29. 99 67.31 20 9. 900 152.18 47. 22 29. 99 6 7.30 AFTER 20 ITERATIONS,THE BEST RETURN IS 9.900 FREE VARIAELES : 152.18 47.22 29.99 67.30 SIEULATE MAIN CROP SI 150 * GROSS * THINNINGS AFTER THINNING * PRODUCTION * 104 AV. TOTAL * TOTAL * AV. TOTAL AGE STEMS B EH EA VOL * VOL MAI * STEMS DBH BA VOL ********************************************* 20 608 4. 3 67. 47 216 11.1 159. 57 120 13.4 131. 67 119 14.9 161. 1101. * 1101. 55 * 5914. * 8472. 179 * 5424. * 10690. 187 * 7164. * 12431. 165 * 156 8.7 8 8 11.3 72. 2558. 67. 2709. OBJECTIVE VALUE IS 10.09 ? STOP # EXECUTION TERMINATED More i t e r a t i o n s (40) of the OPTIMIZE a l g o r i t h m r e s u l t s i n only small refinements i n the p o l i c y and some improvement i n the net present worth. The f i n a l p o l i c y was simulated and the stand t a b l e i s i n c l u d e d i n Appendix VI ( p o l i c y #4). 4.1.5 Test Case: K i l k k i ' s Model The l a s t model to be examined under the o p t i m i z a t i o n s u p e r v i s o r program i s an e m p i r i c a l , d i s t a n c e independent, whole stand model, used by K i l k k i and Vaisonen (1970). The stand model i s l i t t l e more than a s i n g l e volume increment f u n c t i o n but i s i n c l u d e d here as i t was p u b l i s h e d with a complete and d e t a i l e d economic model. Also the simple and continuous nature of the model makes i t i d e a l f o r o p t i m i z a t i o n a n a l y s i s . The s t a t e v a r i a b l e s are the c u r r e n t volume and age of the stand. volume growth f e l l o w s a growth f u n c t i o n c a l c u l a t e d f o r Scots pine (Pinus s j l v e s t r i s 1.) by Kuusela and K i l k k i (1963), t h a t takes the form 105 -r , b , n c v I = at 10 v where I = c u r r e n t volume growth, cu.m./ha/year t = age, years v = volume, cu.m,/ha a,b,c are constants The remainder of the model c o n s i s t s of economic r e l a t i o n s h i p s and assumptions of stand management. A f a m i l y of curves expresses the u n i t value of the growing stock as a f u n c t i o n of age and volume. In c a l c u l a t i o n s concerned with t h i n n i n g from below, the value of growing stock before and a f t e r t h i n n i n g i s c a l c u l a t e d ; the d i f f e r e n c e between these r e p r e s e n t s the value of the volume removed. In t h i n n i n g from above, the value of the growing stock i s assumed to be independent of the d e n s i t y . The value of the cut i s the same as the value of the growing stock and does not change with a l t e r a t i o n s i n the volume removed. Thinning c o s t s and c l e a r c u t t i n g c o s t s are t a b u l a t e d by value of the removals and volume to be cut. K i l k k i and Vaisanen's model was programmed to be com patable with the o p t i m i z a t i o n s u p e r v i s o r . The stand i s grown from an i n i t i a l growing stock l e v e l (cu.m./ha) at age 20 by means of the increment f u n c t i o n . The time and i n t e n s i t y of up to f i v e t h i n n i n g s can be scheduled independently. Only one index of performance i s r e t u r n e d , the present net worth of an acre managed ac c o r d i n g to the s u p p l i e d sequence of a c t i o n s . S e v e r a l m u l t i p l e t h i n n i n g regimes were examined and a two t h i n n i n g regime was chosen to demonstrate the nature of the model and the method of a n a l y s i s . The i n i t i a l p c l i c y had f i v e 106 f r e e v a r i a b l e s ; the age and i n t e n s i t y cf the two t h i n n i n g s , and the age at c l e a r cut. The i n i t i a l growing stock l e v e l at age 20 was assumed to be 6 cu.m./ha and the di s c o u n t r a t e f o r c a l c u l a t i o n of the present net worth was 251. The o p t i m i z a t i o n a l g o r i t h m s converged on the p o l i c y SIMULATEd below: ? SIMULATE VOLUME VOLUME NET AGE BEFORE CUT REMOVED RETURN 55 289.00 122.60 776.70 70 289.00 167.40 1242.90 84 195.00 195.00 1691.08 PRESENT NET WORTH : 1096.84 The simple c o n s t r u c t i o n of the model and r e s u l t a n t f a s t e x e c u t i o n time can be e x p l o i t e d t o examine the pathway c f the c l i m b i n g a l g o r i t h m i n the o p t i m a l neighbourhood. The SUBROUTINE command of the s u p e r v i s o r program was used to l i n k to a p r e v i o u s l y w r i t t e n s u b r o u t i n e t h a t performed 10,000 s i m u l a t i o n s , v a r y i n g the age at f i r s t and second t h i n n i n g between 50 and 60 year s , and 70 and 80 y e a r s , r e s p e c t i v e l y . A l l ether v a r i a b l e s were he l d at t h e i r p r e v i o u s o p t imal values. The r e s u l t i n g s u r f a c e of present net worth was contoured and smoothed (Figure 9). The two t h i n n i n g p o l i c y was then a l t e r e d with the ages at f i r s t and second t h i n n i n g SET t o 58 years and 78 y e a r s , r e s p e c t i v e l y . The value o f the other v a r i a b l e s were Fixed at t h e i r optimum values. 107 Age at second thinning Figure 9 . The pathway of the optimization algorithm on a surface of present net worth. ( K i l k k i ' s model). 108 The pathway followed by the o p t i m i z a t i o n a l g o r i t h m s i s t r a c e d on the contour map (Figure 9) from the i n i t i a l p o l i c y ( i t e r a t i o n 0) to the optimum ( i t e r a t i o n 12). A number of c h a r a c t e r i s t i c s of the s e g u e n t i a l simplex a l g o r i t h m are evident on Figure 9. F i r s t , r a p i d a c c e l e r a t i o n o c c u r s as the a l g o r i t h m s t a r t s e x p l o r i n g about the i n i t i a l point (58,78) with a s m a l l step s i z e (.5, .5) and reaches (55,77) i n t h r e e i t e r a t i o n s and (57. 5,72. 5) by the f o u r t h . The a l g o r i t h m g u i c k l y f i n d s the r i d g e s l o p i n g up from (55,76) towards (55,70), and f o l l o w s the l i n e of s t e e p e s t ascent to the neighbourhood of the optimum. The c o n t r a c t i o n of the simplex near the optimum i s e v i d e n t from the s h r i n k i n g step s i z e at i t e r a t i o n s 4, 5 and 6. The f i n a l stage of e x p l o r a t i o n of the optimal neighbourhood takes 6 more i t e r a t i o n s u n t i l convergence i s reached at i t e r a t i o n 12, The optimum i s j u s t i n s i d e the mapped region a t (55.2,70, 1). F i g u r e 9 a l s o demonstrates the complexity of the o b j e c t i v e s u r f a c e of even a simple and reasonably w e l l behaved model. Some of the i r r e g u l a r i t i e s of the s u r f a c e may be a r t i f a c t s of the c o n t o u r i n g process, but most of the d e t a i l can be a t t r i b u t e d to d i s c r e t e and d i s c o n t i n u o u s elements i n the economic model, such as the t a b u l a r h a r v e s t i n g c o s t s and minimum merchantable volume c o n s t r a i n t s . A l a s t technique of o p t i m i z a t i o n a n a l y s i s to be demonstrated with the o p t i m i z a t i o n s u p e r v i s o r i s parameteric programming. In t h i s type of s e n s i t i v i t y a n a l y s i s , a d e c i s i o n v a r i a b l e i s t r e a t e d as a parameter and v a r i e d over a range of v a l u e s . At s m a l l increments of the parameter, the remaining f r e e 109 d e c i s i o n v a r i a b l e s are allowed t o r e - a d j u s t t o maintain an optimum p o l i c y , Parameteric programming d i f f e r s from the ranging a n a l y s i s d e s c r i b e d i n s e c t i o n 4.1.3 i n that r anging demonstrates the s e n s i t i v i t y of the o b j e c t i v e f u n c t i o n t o changes i n one d e c i s i o n v a r i a b l e , with a l l other v a r i a b l e s held c onstant. Parameteric a n a l y s i s demonstrates the s e n s i t i v i t y of the optimum o b j e c t i v e value to changes i n one d e c i s i o n v a r i a b l e . Parameteric programming may be accomplished with the o p t i m i z a t i o n s u p e r v i s o r by using the FIX command to remove a v a r i a b l e from c e n t r e ! of the o p t i m i z a t i o n a l g o r i t h m , and repeated a p p l i c a t i o n of the SET and OPTIMIZE command. A parametric type of s e n s i t i v i t y a n a l y s i s was performed on the age and i n t e n s i t y of the f i r s t t h i n n i n g . In both cases the d e c i s i o n v a r i a b l e t r e a t e d as a parameter i s f i x e d , and the remaining f r e e v a r i a b l e s take cn optimum va l u e s . The d e c i s i o n v a r i a b l e s o f age and i n t e n s i t y of i n i t i a l and subsequent t h i n n i n g s , and age at c l e a r c u t , are numbered 1-4 and 11, r e s p e c t i v e l y . When the i n t e n s i t y of the f i r s t t h i n n i n g i s t r e a t e d as a parameter and incremented from 100 to 175 cu.m./ha, the remaining f r e e v a r i a b l e s i n the optimum p o l i c y f l u c t u a t e i n response, i n a smooth and continuous manner (Table 3). There are no r a d i c a l d i s c o n t i n u i t i e s i n the op t i m a l p o l i c y space, such as might r e s u l t , f o r example, from the e l i m i n a t i o n of a t h i n n i n g . As the i n t e n s i t y of the i n i t i a l t h i n n i n g i n c r e a s e s , the age a t the f i r s t and second t h i n n i n g i n c r e a s e s , while the i n t e n s i t y of the second t h i n n i n g decreases. The age at c l e a r c u t f l u c t u a t e s only s l i g h t l y . S i m i l a r l y , when the age at f i r s t t h i n n i n g i s t r e a t e d as a 110 Table 3, S e n s i t i v i t y of the optimal two t h i n n i n g management seguence t o v a r i a t i o n s i n the i n t e n s i t y of the f i r s t t h i n n i n g . ( K i l k k i ' s model) V 2 MAX: FREE VARIABLES (Fixed) M I 1 1 3 4 1 1 100.0 1094.9 53.8 69.7 187.6 83. 3 105. 0 1095. 0 54. 1 69.6 184. 6 84.5 110.0 1095.3 54.6 69.3 181.3 84. 1 115.0 1096.7 54.9 70. 2 174. 5 84. 1 117.5 1096.7 54. 9 70. 2 170. 7 84. 1 120.0 1096.7 54.9 70. 2 170. 6 83. 1 122.62 1096.8 54.9 70.2 167. 8 84. 1 125. 0 1096.6 54. 9 70. 2 166. 9 84. 1 130.0 1096. 0 54.9 70.2 166.9 84. 1 135. 0 109 5. 2 55. 9 70. 2 166. 9 84.1 145.0 1094.8 55.9 72. 1 166.9 84.1 155.0 1092.1 57.0 72. 7 166. 4 84.4 165. 0 1090.3 57.3 76.8 160. 0 85.4 175.0 1089.9 57. 1 77. 1 160. 8 85.4 1 Present net worth (PNW) i n d o l l a r s i s c a l c u l a t e d with a di s c o u n t r a t e of 2%, 2 The g l o b a l optimum. parameter and v a r i e d between 50 and 60 years, the trends i n the f r e e v a r i a b l e s show no r a d i c a l changes i n p o l i c y (Table 4 ) . The i n t e n s i t y of the f i r s t t h i n n i n g i n c r e a s e s markedly, while the ages at the second t h i n n i n g and c l e a r c u t change only s l i q h t l y . The i n t e n s i t y of the second t h i n n i n g shews a downward t r e n d with i n c r e a s i n g age a t f i r s t t h i n n i n g . Of course, a parametric a n a l y s i s may be performed with a model parameter as w e l l as d e c i s i o n v a r i a b l e s . Table 5 re c o r d s the optimal two t h i n n i n g p o l i c i e s as the d i s c c u n t r a t e i n c r e a s e s . As the discount r a t e r i s e s , the age at f i r s t t h i n n i n g i s reduced and the i n t e n s i t y of the f i r s t t h i n n i n g i s i n c r e a s e d . 111 Table 4. S e n s i t i v i t y o f the o p t i m a l 2 t h i n n i n g management seguence to v a r i a t i o n i n the age of the f i r s t t h i n n i n g ( K i l k k i 's model). V 1 KAX: FREE VARIAELES (Fixed) 111 4 2 3 4 1 1 50 10 82.5 101.0 69.5 185.8 85.0 51 1084.6 112.0 70.5 177. 9 85.8 52 10 93.3 116.7 71. 0 173. 8 84.8 53 10 94.0 117.5 70. 6 170. 8 85.7 54 1097.4 118.2 70. 0 172.4 85.2 54.92 1096.8 122. 7 70. 2 167. 2 84.1 55 1096.9 123.0 70.2 167. 1 84.0 56 10 93.3 129.2 64. 9 165. 6 84.0 57 10 93.0 132.0 7C.3 165. 8 84.2 58 1091.7 132. 1 70. 3 170. 8 84.2 59 1088.0 135.6 70. 3 169.3 84.0 60 1084.1 136.5 71. 4 168. 6 84.4 1 Present net worth (PNI) i n d o l l a r s i s c a l c u l a t e d with a d i s c o u n t r a t e of 2%. 2 The g l o b a l optimum. At i n t e r e s t r a t e s above 4$, the f i r s t t h i n n i n g i s e l i m i n a t e d . 4.1,6 Summary And D i s c u s s i o n The f o r e g o i n g examples have demonstrated t h a t f o r e s t stand s i m u l a t i o n models t h a t are to be subjected to o p t i m i z a t i o n a n a l y s i s , should have the f o l l o w i n g c h a r a c t e r i s t i c s : f a s t e xecuting -The s i m u l a t i o n model should be as simple as p o s s i b l e , 112 Optimum management seguences with two commercial t h i n n i n g s , at i n c r e a s i n g d i s c o u n t r a t e s , ( K i l k k i ' s model) M i l MAX: FREE VARIABLES PNJi 3 4 U .01 3246.4 62 102 74 160 92 .02 1096.9 55 122 70 167 84 . 03 498.9 53 138 69 165 83 . 04 257. e 48 144 69 169 84 . 04 + two commercial t h i n n i n g s are u n p r o f i t a b l e 1 Present net worth (PKH) i n d o l l a r s i s c a l c u l a t e d with a di s c o u n t r a t e c f 2% . w i t h i n the c o n s t r a i n t s of v a l i d i t y and d e s i r e d r e s o l u t i o n , so t h a t e x e c u t i o n time i s minimized. Nelder and Mead (1965) reported t h a t the number of e v a l u a t i o n s of the o b j e c t i v e f u n c t i o n needed f o r the s e q u e n t i a l search t o converge i s p r o p o r t i o n a l t o the sguare of the number of f r e e v a r i a b l e s . d e t e r m i n i s t i c -S t o c h a s t i c elements add to the e f f o r t needed to rank p o l i c y p o i n t s on the o b j e c t i v e s u r f a c e , without adding much i n f o r m a t i o n to a d e c i s i o n a n a l y s i s . S t o c h a s t i c elements should be avoided wherever p o s s i b l e . unimodal -Multimodal o b j e c t i v e s u r f a c e s ' p r o b a b l y can't be avoided, but care should be taken not to c r e a t e multimodal Table 5. 113 s i t u a t i o n s with pe n a l t y c o n s t r a i n t s or aggregated v a r i a b l e s . continuous -D i s c o n t i n u i t i e s i n the o b j e c t i v e s u r f a c e can be caused by d i s c r e t e elements i n the model, such as t a b u l a r data or c o n s t r a i n t s , or d i s c o n t i n u o u s f u n c t i o n a l r e l a t i o n s h i p s . These d i s c o n t i n u i t i e s reduce the e f f i c i e n c y of the o p t i m i z a t i o n a l g o r i t h m , and may cause f a l s e convergence. responsive -One should avoid c r e a t i n g d e c i s i o n v a r i a b l e s t h a t aggregate other independent and c o u n t e r a c t i n g v a r i a b l e s . For example, to reduce the s i z e of the d e c i s i o n space of Goulding's model, a s i n g l e d e c i s i o n v a r i a b l e c o n t r o l l e d the i n t e n s i t y of a l l t h i n n i n g s . I t i s easy to imagine a s i t u a t i o n where i t would be advantageous to decrease the i n t e n s i t y of an e a r l y t h i n n i n g and i n c r e a s e the i n t e n s i t y of a l a t e r c u t . The examples a l s o show the value and n e c e s s i t y of a h i g h l y i n t e r a c t i v e environment f o r t h i s kind of a n a l y s i s . Attempts to use the system i n batch mode, without i n t e l l i g e n t i n t e r v e n t i o n , have been u n s u c c e s s f u l . A major r a m i f i c a t i o n of t h i s need f o r i n t e r v e n t i o n i s that d i r e c t c l i m b i n g i s not s u i t a b l e f o r o p t i m i z i n g a submodel under c o n t r o l of an automatic process, such as the decomposition approach suggested i n S e c t i o n 3.4.4. 114 4.2 ME2 As A Network Problem Se c t i o n 3.4.1 concluded with the statement t h a t MP2, the d e c i s i o n problem of f i n d i n g the o p t i m a l management sequence f o r a treatment u n i t , can be formulated as an i n i t i a l value problem and s o l v e d by dynamic programming. In t h i s s e c t i o n , MP2 w i l l be formulated as a network problem t h a t i s e q u i v a l e n t to the i n i t i a l value problem. The dynamic programming s o l u t i o n a l g o r i t h m e i t h e r a s s i g n s c o s t s and b e n e f i t s to each a r c , or r e f e r e n c e s an a p p r o p r i a t e model to simulate the t r a n s i t i o n . In order to formulate the s e t A u of a l l management sequences f e a s i b l e cn treatment u n i t u, as a network, the n a t u r a l l y continuous s t a t e v a r i a b l e s of the treatment u n i t must be made d i s c r e e t . T h i s process i s best d e s c r i b e d with a simple example. Consider a Scots pine stand, where qrowth and y i e l d can be p r e d i c t e d by K i l k k i ' s model, d e s c r i b e d e a r l i e r . I t i s d e s i r e d that the s e t of manaqement sequences i n c l u d e the a l t e r n a t i v e s of n a t u r a l r e g e n e r a t i o n or p l a n t i n g t o two d i f f e r e n t s t o c k i n g l e v e l s , and t h r e e d i f f e r e n t i n t e n s i t i e s of t h i n n i n g . The stand i s c u r r e n t l y overmature, i n a w i l d s t a t e , and the f i n a l harvest w i l l be a c l e a r c u t . Curves r e p r e s e n t i n g volume (cu.m./ha.) are generated from nine d e n s i t y l e v e l s at age 20 years (6., 4., 3., 2.5, 2., 1.5, 1., .75, .5 cu.m./ha.) using K i l k k i ' s model. The curves w i l l be i d e n t i f i e d by t h e i r d e n s i t y l e v e l (Dl) at age 20 years (Figure 10). D i s c r e t e s t a t e s corresponding to the management a c t i o n s can now be d e s c r i b e d i n terms of the DI c u r v e s . In the example, we w i l l assume t h a t the ' a c t i v i t y ' of n a t u r a l r e g e n e r a t i o n w i l l r e s u l t i n the stand being i n the 115 IT ) . o i n . cn in. m 5 Q . o a — CD. Qj CM E O a a. a. in / t , J , f , y j j / / / J / ' ' • •> j' / / t t , y" D L = 6 . 0 / / f t , t / , , , / ' 4 . 0 ' J ' ' t J , , , ' < 3.0 y ' ; t t < 2S ' / / / / / / ' 1-0/ f ' ' ' ' • / ••/ -75/ J . l i t ' ' -t , / r J , ' / ' 0 , 5 ' ' ' ' 1 J / ' / / / / / • ' / ' • J i i > ' i i , i i t i i i > t i i ' ' ' . i i curve / / j , j ' ' f j ' t 9' / //•/ < / < / '//// / > ' <' 8 / / / > / • 7 ' ' ' t ' , 5 ' / ' -4 ' ^ -> y J " 2 - * -1 " 1 I 3 0 . 0 4 0 . 0 5 0 . 0 BO.D A g e ( y e a r s ) 7 0 . 0 8 0 . 0 S O . O 1 D D . 0 Figure 10. Volume/age curves corresponding to 9 lev e l s of density at age 20 years, of Scots pine ( K i l k k i ' s model). 116 • s t a t e ' of having a d e n s i t y of 3 cu.m./ha. at age 20 years, while p l a n t i n g w i l l r e s u l t i n a s t a t e of DL = 4 or 2 cu.m./ha. A n a t u r a l l y regenerated stand can be thinned only to the s t a t e of DL = 2 cu.m./ha. The s t a t e of the c u r r e n t wild stand i s i t s c u r r e n t volume, and i s not expected t o change before h a r v e s t . The s t a t e r e s u l t i n g from a c l e a r c u t i s , of course, 0.0 cu.m./ha. These s t a t e s are l i s t e d i n F i g u r e 11, accompanied by a d i r e c t e d c y c l i c graph t h a t r e p r e s e n t s the precedence r e l a t i o n s h i p s and f e a s i b l e changes of s t a t e . The precedence graph (Figure 11) r e p r e s e n t s a s e t of a l t e r n a t i v e management sequences i n terms of c h o i c e of management a c t i o n s and t h e i r i n t e n s i t i e s , but does not i n c l u d e any i n f o r m a t i o n p e r t a i n i n g to the t i m i n g of the management a c t i o n s . For example, the c u r r e n t wild stand may be cut any time xn the f i r s t decade, p r o v i d i n g i t i s c l e a r c u t by the end of the decade. The f i r s t and l a s t e n t r y times f o r each s t a t e and a time staged r e p r e s e n t a t i o n o f the precedence graph i s given i n F i g u r e 12. The ar c s i n the time staged graph r e p r e s e n t s t a t e t r a n s i t i o n s t h a t are f e a s i b l e at a s p e c i f i c stage. A l l stages i n the example problem are c f egual l e n g t h , f i v e years. The c y c l i c nature of the precedence graph i s not represented d i r e c t l y but i s s i m p l i f i e d to the assumption t h a t the management seguence f o r the regenerated stand w i l l be repeated i n f i n i t e l y . A l l a r c s are assumed t o be d i r e c t e d , l e f t to r i g h t . The c h o i c e of an arc r e p r e s e n t s a management a c t i o n at a st a g e , so a management sequence can be thcuqht of as any pathway through the graph from the source node S, to the sink F. The set of a l l f e a s i b l e management seguences i s the set of a l l pathways throuqh F i g u r e 1 2 . A l t e r n a t i v e m a n a g e m e n t s e q u e n c e s r e p r e s e n t e d a s a c y c l i c d i r e c t e d g r a p h o f s t a t e s a n d s t a g e s . 118 State Des c r i p t i o n 1 wild stand 2 c l e a r c u t 7 natural regeneration, curve 7 9 p l a n t i n g , curve 9 3 thinning cut, curve 8 5 thinning cut, curve 5 1 0 c l e a r c u t Figure 1 1 . A l t e r n a t i v e management sequences represented as a c y c l i c d i r e c t e d graph. the graph from S'to F. The g r a p h i c a l approach i s an e f f i c i e n t s t r u c t u r e t o s t o r e a l a r g e number of a l t e r n a t i v e management sequences. The t o t a l number of pathways through t h e ; graph may be e a s i l y counted. S t a r t i n g with the source node S, the v e r t i c e s of the graph are l a b e l l e d with the t o t a l number of pathways back to the source node. For each a r c i n c i d e n t at a given vertex, the number of pathways back to S i s the l a b e l of the vertex t h a t the a r c came from. The l a b e l s a s s o c i a t e d with each a r c i n c i d e n t at the vert e x are summed t o form the vertex l a b e l . The l a b e l computed f o r the 119 s i n k node F i s the t o t a l number of pathways through the graph. The counting a l g o r i t h m can be expressed i n the n o t a t i o n of S e c t i o n 3.3.2 and w i l l help e l u c i d a t e the r e l a t i o n s h i p between the a c t i v i t y graph and the m u l t i s t a g e f o r m u l a t i o n of MP2. The s u b s c r i p t corresponding to the treatment u n i t u can be dropped to s i m p l i f y the n o t a t i o n , as i t i s understood we are c o n s i d e r i n g a s i n g l e treatment u n i t of Scots pine. As before, the s t a t e on i n p u t t o stage t i s v and on output i s v . V i s the set of d i s c r e t e s t a t e s that are input to stage t . Each vertex of the graph at the beginning of stage t corresponds to a s t a t e v . The s t a t e t r a n s i t i o n f u n c t i o n Eg. (3.3-18) with the chosen management a c t i o n can be thought of as the a r c l i n k i n g the two d i s c r e t e s t a t e s v f c and v : M I i (4.2-1) v t = M ( v t , a t ) performing s t a t e i n v e r s i o n on Eg. (4.2-1) and using the i n c i d e n c e i d e n t i t y Eg. (3.3-19), provides an e x p r e s s i o n f o r the i n p u t s t a t e s c f a r c s i n c i d e n t on a given v e r t e x v ; Let n(v f c) be the number of pathways from the source node S to v t < Then n (F) can be found r e c u r s i v e l y from the r e l a t i o n s h i p n ( v ) = E n ( v ) = E n ( M ( v , a ) ) (4.2-3) 120 f o r a l l v V and t = 1, 2, . .., T. The management seguence graph f o r the demonstration problem (Figure 12) has been l a b e l l e d using t h i s procedure. Even f o r t h i s simple problem, there are 63 a l t e r n a t i v e management seguences. I f the management a c t i o n r e t u r n s R t ( v t , a t ) are placed on the a r c s , then the MP2 problem of f i n d i n g the optimum management seguence i s e g u i v a l e n t to f i n d i n g the optimum pathway from S to F, Formulated as a network m o d e l 1 0 , the dynamic programming s o l u t i o n d e s c r i b e d i n s e c t i o n 3.4,1 can be used to f i n d the optimal path. Using the s t a t e i n v e r s e form of the t r a n s i t i o n f u n c t i o n (Eg, 4.2-1) allows the s o l u t i o n procedure to proceed i n a forward manner from t = 1, T. The management a c t i o n r e t u r n E (v ,a ) can be made a f u n c t i o n of the output s t a t e v with the i n v e r s e t r a n s i t i o n f u n c t i o n : R" (v t,a f c) = R ( M ' (v t,a t)) (4.2-4) In t h i s form, the management a c t i o n r e t u r n can be i n t e r p r e t e d as the c o s t or p r o f i t of e n t e r i n g s t a t e v; v i a management a c t i o n (arc) afc , For stage t=1, there i s only one a r c from S to v , and the 1 0 A graph d e f i n e s the purely s t r u c t u r a l r e l a t i o n s h i p s between the nodes, while a network bears a l s o the g u a n t i t a t i v e c h a r a c t e r i s t i c s of the nodes and a r c s . For a c o n c i s e d e s c r i p t i o n of network models see Elmaqhraty (1970). 121 maximum stage r e t u r n i s f l ( V - V W ,4.2-5) f o r a l l v - ^ v 2 . For stages t = 2, ..,, T, the usual dynamic programming r e c u r s i o n takes the form: f t(v t) = M a x i m u m + W V V (4.2-6) where f t - i ( v t ' a t ) = f t - i ( M V i ' V 1 (4.2-7) At the f i n a l stage v T = F, and the o p t i m a l pathway a T , a , * . .., a 1 can be t r a c e d back through the optimal stage r e t u r n s . The network f o r m u l a t i o n of MP2 i s e a s i l y solved by forward r e c u r s i o n , as s t a t e i n v e r s i o n of the t r a n s i t i o n f u n c t i o n i s s i m p l i f i e d to t r a c i n g the arc backwards r a t h e r than forwards through a stage. However, i n the more general f o r m u l a t i o n of MP2 d e s c r i b e d i n S e c t i o n 3,3.2, s t a t e i n v e r s i o n would i n v o l v e s o l v i n g a complicated s t a t e t r a n s i t i o n f u n c t i o n or s i m u l a t i o n model f o r an input s t a t e as a f u n c t i o n cf the output s t a t e and management a c t i o n . 122 4.2.1 Generating The Graph Of A l t e r n a t i v e Management Seguences The network f o r m u l a t i o n of MP2 i s c o n c e p t u a l l y s i m p l e , yet f l e x i b l e and e f f i c i e n t enough t o encompass l a r g e s e t s of a l t e r n a t i v e management seguences. Of course, f o r m u l a t i n g a management a l t e r n a t i v e graph and adding the a r c r e t u r n s i s extremely t e d i o u s , i f not i m p o s s i b l e , f o r any but the s i m p l e s t problems, To make t h i s approach c o m p u t a t i o n a l l y f e a s i b l e , a program was w r i t t e n t h a t generates a graph encoding the s t r u c t u r a l c h a r a c t e r i s t i c s o f the management seguences, from a simple o u t l i n e of the management problem. The q u a n t i t a t i v e aspects of the problem, the c o s t s and p r o f i t s d e f i n e d on the a r c s , can be assigned d u r i n g the generation s t e p , but are u s u a l l y computed as needed during the l a t e r s o l u t i o n step. Through ge n e r a t i n g the s t r u c t u r a l aspects of the problem as a separate s t e p , the r e s u l t i n g management sequence graph can be c a l i b r a t e d with arc r e t u r n s f o r d i f f e r e n t treatment u n i t s or c o s t b e n e f i t assumptions at s o l u t i o n time, r a t h e r than r e g e n e r a t i n g and s t o r i n g a network f o r each v a r i a t i o n on the problem. The procedure f o l l o w e d by the graph generating program i s d e s c r i b e d below: 1 : Read i n and set up the problem time frame. The graph generating program allows a user to d e f i n e a v a r i a b l e time frame f o r the problem, so t h a t each stage can represent a d i f f e r e n t time span. For example, i f a stand i s to be n a t u r a l l y regenerated and grown without d i s t u r b a n c e f o r 40 y e a r s , a s i n g l e stage of 40 years 12 3 c o u l d cover t h i s time span. The f o l l o w i n g stages might rep r e s e n t 2-year i n t e r v a l s , to provide s u f f i c i e n t r e s o l u t i o n f o r s c h e d u l i n g a t h i n n i n g a c t i o n . T h i s completely f l e x i b l e time frame means t h a t problem s o l v i n g e f f o r t can be concentrated i n those time p e r i o d s t h a t are of most i n t e r e s t , r a t h e r than being evenly a l l o c a t e d over the planning h o r i z o n . Read i n a management o u t l i n e f o r a treatment u n i t . The management o u t l i n e c o n s i s t s of a d e s c r i p t i o n o f the s t r u c t u r a l aspects of the management seguence graph and a parameter l i s t d e f i n i n g the treatment u n i t s i m u l a t i o n model to be used i n s i m u l a t i n g the s t a t e t r a n s i t i o n s . For each s t a t e , the graph d e s c r i p t i o n c o n s i s t s o f : - the e a r l i e s t and l a t e s t time period t h a t the s t a t e can be entered, - the minimum and maximum d u r a t i o n i n years of the t r a n s i t i o n i n t o the s t a t e , - the c o s t s per u n i t area or per u n i t volume of e n t e r i n g the s t a t e (revenues are t r e a t e d as negative c o s t s ) , - the wood volume produced on e n t e r i n g the s t a t e , - a f l a g t o s i g n i f y whether a d d i t i o n a l c o s t or revenue i n f o r m a t i o n i s to be generated at s o l u t i o n time by s i m u l a t i n g the t r a n s i t i o n i n t o the s t a t e , - a l i s t of output s t a t e s that may be entered d i r e c t l y from the c u r r e n t s t a t e . Generate the graph, Arcs are generated as the 4-tuple (from stage, from s t a t e , at stage, at state) (Figure 13) and s o r t e d i n t o ascending order by stage and s t a t e to 124 f a c i l i t a t e the dynamic programming s o l u t i o n process. 4 : The a r c s are added to a l i s t . I f there are more treatment ' f r o m ' states output s tates t ' f r om ' stage t+1 ' a t ' stage Figure 13. L a b e l l i n g of arcs produced by the graph generating program. u n i t s , s t e p 2 through 4 are repeated. Otherwise proceed to step 5. 5 : The graph l i s t i s saved on a permanent f i l e . A t e s t problem was c o n s t r u c t e d s i m i l a r t o K i l k k i ' s (1970) i n v e s t i g a t i o n of t h i n n i n g s t r a t e g i e s . Stands of Scots pine were assumed to be 50 years o l d , grown without i n t e r v e n t i o n from stands of 6 cu.m./ha. At age 20 years. The d e c i s i o n problem was to schedule the t i m i n g and i n t e n s i t y of up to three t h i n n i n g s , f o l l o w e d by a c l e a r c u t . The problem s t a t e s , d e f i n e d i n terms of stand d e n s i t y l e v e l s a t age 20, are l i s t e d i n F i g u r e 14, together with a precedence graph showing the p o s s i b l e combinations of t h i n n i n g s and a c l e a r c u t . 125 c c —-State Description 1 DL = .5 cu.m./ha. at age 20 2 .75 3 " 1.00 4 " 1.50 5 " 2.00 6 " 2.50 7 " 3 . 0 0 8 " 4.00 9 " 6.00 10 cle a r c u t 11 i n i t i a l (dummy) state Figure 14. State d e f i n i t i o n and precedence graph of a multiple thinning decision problem ( K i l l k i ' s model). 126 Quite complex t h i n n i n g s t r a t e g i e s are encoded i n the precedence graph. From the i n i t i a l s t a t e 9 (Dl=6), t h i n n i n g s are p o s s i b l e to s t a t e s 7 (DL=3) , s t a t e 6 (DL=2.5), and s t a t e 5 (DL=2). Second t h i n n i n g s are p o s s i b l e from s t a t e 7 and 6 to s t a t e s 4, 3, 2, and 1 (Dl=1,5, 1.0, .75, .5, r e s p e c t i v e l y ) . However, from s t a t e 5, a second t h i n n i n g can only r e s u l t i n s t a t e s 3, 2 or 1. S i m i l a r l y , subseguent t h i n n i n g s from s t a t e 4 can r e s u l t i n s t a t e 2 or 1, but a t h i n n i n g from s t a t e 3 can only be to s t a t e 1. The c l e a r c u t , s t a t e 10, i s a c c e s s i b l e from a l l t h i n n i n g s t a t e s . In a d d i t i o n to the s t a t e precedences, i n f o r m a t i o n on t i m i n g of the s t a t e t r a n s i t i o n s i s necessary to generate the a c t i v i t y graph. The i n i t i a l t h i n n i n g s t a t e s (7, 6, and 5) may be entered between year 51 and year 71. Any subseguent t h i n n i n g s must be accomplished by year 86. State 4 may be entered at year 61, s t a t e 3 at year 63, s t a t e 2 at year 65, and s t a t e 1 at year 67. A c l e a r c u t may not occur before year 76 but must be done before year 91. The program e x e c u t i o n command and the management o u t l i n e data s e t are l i s t e d i n Appendix VII. The time frame d e f i n e d f o r the graph c o n s i s t e d of an i n i t i a l stage of 1 year f o l l o w e d by a second stage of 51 y e a r s . Subsequent stages were a l l of 2 years d u r a t i o n . As no management a c t i o n s were t o take place i n the f i r s t 50 years of the s tands' growth, t h i s time frame concentrated r e s o l u t i o n and s o l u t i o n e f f o r t d u r i n g the p e r i o d of i n t e n s i v e management. A f t e r generating the graph to a permanent f i l e , a separate program was used to d i s p l a y i n f o r m a t i o n about the graph. F i g u r e 127 15 i s a management o u t l i n e r e p o r t and i s simply a formatted l i s t i n g of the management o u t l i n e data s e t , d e s c r i b e d above. 1 M M M M 1 M i l ! J : SCOTS P I N E , AGE 50 E N T R Y STATE FIRST LAST DURATION MIN MAX COSTS AREA VOL 15 | | j I | F ! 9 9 ! 41 41 | | I | T ! 7 6 5 10 1 1 53 71 | | I | T ! 4 3 2 1 6 I —• w 71 J | I | T ! 4 3 2 1 5 I 53 71 | | I I T ! 3 2 1 10 4 i 61 86 | \ I | T ! 2 1 10 3 } 6 3 86 | | I | T | 1 10 2 ! 65 86 | | I | T | 10 1 I 67 86 | | 1 J T | 10 10 J 76 91 | | 1 1 T | 10 ! 93 I I 1 | F | VOLUME CUT SIM OUTPUT STATES 10 THE CORRESPONDING NETWORK HAS 29886 FEASIBLE MANAGEMENT SEQUENCES Fi g u r e 15. A management o u t l i n e r e p o r t f o r m u l t i p l e t h i n n i n g s of Scots pine. Minimum and maximum d u r a t i o n times of t r a n s i t i o n s i n t o a s t a t e may be s p e c i f i e d , and the graph program w i l l generate a r c s of the a p p r o p r i a t e stage l e n g t h . In the demonstration problem, t h i s o p t i o n i s not used and i t i s assumed t h a t a l l t r a n s i t i o n s occur d u r i n g one time stage (two y e a r s ) . The s i m u l a t i o n f l a g (SIM FLAG) set to t r u e (T) f o r any s t a t e s i g n i f i e s t h a t the t r a n s i t i o n i n t o the s t a t e i s t o be simulated with the 128 a p p r o p r i a t e model, to provide c o s t and b e n e f i t i n f o r m a t i o n . The r e p o r t program uses the counting a l g o r i t h m d e s c r i b e d i n S e c t i o n 4.2.1 and c a l c u l a t e d 29886 unigue pathways ( f e a s i b l e management seguences) through the a c t i v i t y graph. A p a r t i a l l i s t i n g of the a r c s i n the graph i s i n c l u d e d i n Appendix VII. The r e p o r t program w i l l a l s o generate a p l o t of the a c t i v i t y graph (e.g. F i g u r e 12), which can be used to check f o r l o g i c a l i n c o n s i s t e n c i e s i n the generated graph. However, the p l o t of the graph demonstration problem was too dense to be i n c l u d e d here. A s i m p l i f i e d v e r s i o n of the generated graph i s represented i n Figure 16. The a d d i t i o n of 'no change' a r c s (e.g. from s t a t e 9 to s t a t e 9 d u r i n g stage 2) c o r r e c t l y r e p r e s e n t s the pathway through the graph but decreases the apparent number of a r c s . The network problem cannot be c o r r e c t l y solved i n t h i s form. 4.2.2 F i n d i n g The Optimum Management Sequence The computer program t h a t s o l v e s the network problem g e n e r a l l y f o l l o w s the dynamic programming a l g o r i t h m d e s c r i b e d i n S e c t i o n 4.2,1. The program SOLVE reads i n the graph f i l e as a l i s t of a r c s . The a r c s are ordered so t h a t the s o l u t i o n a l g o r i t h m can process the a r c s s e q u e n t i a l l y i n the arc l i s t , on a s i n g l e pass. The computational burden of d e a l i n g with precedence r e l a t i o n s h i p s t h a t would normally be a s s o c i a t e d with a network program, has been t r a n s f e r r e d to the graph g e n e r a t i n g program. F i g u r e 1 6 . A g r a p h r e p r e s e n t i n g t h e a l t e r n a t i v e m a n a g e m e n t s e q u e n c e s o f a m u l t i p l e t h i n n i n g p r o b l e m i n S c o t s p i n e 130 Consequently the s o l u t i o n process i s f a s t and i n e x p e n s i v e ($1,19 f o r the t e s t problem i n v o l v i n g 29,886 a l t e r n a t i v e management sequences) , The s o l u t i o n program i n t e r p r e t s the nature of the t r a n s i t i o n represented by an a r c , and checks i f a s i m u l a t i o n i s r e g u i r e d to provide c o s t / b e n e f i t i n f o r m a t i o n . I f necessary, the a p p r o p r i a t e treatment u n i t model i s executed to estimate the r e t u r n on the s t a t e t r a n s i t i o n . Consequently, the same graph can be entered with d i f f e r e n t s e t t i n g s of model parameters such as s i t e index or d i s c c u n t r a t e . A runstream f o r the t e s t problem i s i n c l u d e d i n Appendix VII. A c o n t r o l card a l l o w s the user t o a s s i g n the treatment u n i t model, c o n t r o l the number and nature of output r e p o r t s , s e l e c t an o b j e c t i v e f u n c t i o n (e.g. maximize PNw) , a s s i g n a discount r a t e , and s p e c i f y a r e g e n e r a t i o n l a g . Examples of the s o l u t i o n program r e p o r t s are i n c l u d e d i n Appendix V I I , f o r the t e s t problem. The r e s u l t s are d i s p l a y e d g r a p h i c a l l y i n F i g u r e 17. with the o b j e c t i v e of maximizing net present worth with a d i s c o u n t r a t e of 2%, t h i n n i n g s were scheduled dur i n g years 54-56 and 72-74. The volumes cut were 129 and 158 cu.m./ha., r e s p e c t i v e l y . A f i n a l h a r v e s t cut was scheduled during years 82-84 and y i e l d e d 193 cu.m./ha. The box on F i g u r e 17 r e p r e s e n t s the time c o n s t r a i n t s f o r e n t e r i n g a given s t a t e . Note that the optimum management sequence i s an i n t e r i o r optimum, with no c o n s t r a i n t s ' t i g h t ' . The o p t i m a l i t y of the management seguence was subseguently t e s t e d with the s i m u l a t i o n o p t i m i z a t i o n s u p e r v i s o r program (SIMOPT) d e s c r i b e d i n S e c t i o n 4.1. A p o l i c y f i l e c o n t a i n i n g the 131 Figure 17. n optimal management sequence c a l c u l a t e d by embedding K i l k k i ' s Scots pine model i n a network formulation. 132 management seguence s e l e c t e d by the network s o l u t i o n program was c r e a t e d with the management a c t i o n s o c c u r r i n g a t the midpoint of the time stages. For example, the f i r s t t h i n n i n g was scheduled to occur i n the stage corresponding t o years 54-56. For the s i m u l a t i o n o p t i m i z a t i o n p o l i c y , the f i r s t t h i n n i n g was assumed to occur at the midpoint of the st a g e , 55 y e a r s . The p o l i c y was simulated and optimized i n the user s e s s i o n recorded below; ? READ KKDF 4 POLICY KKDP , ? SIMULATE 7 PARAMETERS VOLUME VOLUME AGE BEFORE CUT REMOVED 5 VARIABLES ARE READ IN. NET RETURN 55 73 83 279.OC 298. 00 193.00 129.00 158.00 193.00 813.94 1198.79 1661.25 PRESENT NET WORTH : 1071.89 OBJECTIVE VALUE IS 1071.89 ? OPTIMIZE 20 V A R I A B L E S # Z* 1 3 11 0 1071.891 55. 0 0 73. 00 83.00 1 1072.906 54.50 73.00 83.00 2 1072.906 54.50 73. 00 83. 00 3 1072.906 54.5 0 73.00 83. 00 4 1072.906 54.50 73. 00 83. 00 5 1073.575 54. 23 72. 89 84.60 6 1073.575 54.23 72. 89 84. 60 7 1073.57 5 54.23 72. 89 84. 60 8 1074. 574 53.93 73. 15 83. 59 9 1074.574 53. 93 73. 15 83.59 10 1074. 574 53. 93 73. 15 83. 59 11 1074.574 53.93 73. 15 83. 59 12 107 4. 574 53. 93 73. 15 83. 59 13 107 4.574 53. 93 73. 15 83.59 14 1074.998 53. 94 73. 17 83. 54 15 1074.998 53.94 73. 17 83.54 16 1074.998 53. 94 73. 17 83. 54 133 17 1075.108 53.93 73.16 83.53 18 1075. 108 53. 93 73. 16 83. 53 19 1075.108 53.93 73.16 83.53 20 1075.108 53.93 73.16 83.53 AFTER 20 ITERATIONS,THE BEST RETURN IS 1075.108 FREE VARIAELES : 53. 93 73. 16 83. 53 ? STOP When SIMULATEd through SIHOPT , the p o l i c y has a net present worth s l i g h t l y l e s s than that r e t u r n e d by the network s o l u t i o n program ($107-1.89 vs. $1072. 3) . This i s probably due to s m a l l d i f f e r e n c e s i n the computational procedure. The OPTIMIZATION command r e f i n e s the p o l i c y only s l i g h t l y , and i n c r e a s e s the present net worth t c $1075.11. Of course, convergence t o the same p o l i c y does not prove the g l o b a l o p t i m a l i t y of the p o l i c y or confirm the v a l i d i t y of e i t h e r approach to o p t i m i z a t i o n . However, c o n s i s t e n c y i s , at l e a s t , encouraging, A l a s t e x e r c i s e attempted with the network f o r m u l a t i o n was the demonstration of the s e n s i t i v i t y of the o p t i m a l p o l i c y to changes i n the d i s c c u n t r a t e , A second t e s t problem i n v o l v i n g K i l k k i ' s model, very s i m i l a r t o the previous m u l t i p l e t h i n n i n g d e c i s i o n problem, i s d e s c r i b e d i n Appendix VII. The management o u t l i n e and s o l u t i o n runstream are a l s o i n c l u d e d i n the appendix. The optimal management seguences corresponding to di s c o u n t r a t e s of 2%, 3%, . . . . . . 8% are recorded on F i g u r e 18. As one would expect, the second t h i n n i n g and har v e s t c u t s are scheduled at younger ages as the discount r a t e r i s e s . At r a t e s 1% and 8%, the second t h i n n i n g i s e l i m i n a t e d completely and the stand i s c l e a r c u t at 66 years. 1 3 4 CD 3 0 . 0 40.0 5 0 . 0 6 0 . 0 7 0 . 0 2 0 . 0 S O . O 1 0 0 . 0 Age (years) F i g u r e 1 8 . S e n s i t i v i t y o f t h e o p t i m a l m a n a g e m e n t s e q u e n c e t o d i f f e r e n t d i s c o u n t r a t e s ( K i l k k i ' s m o d e l i n t h e n e t w o r k f o r m u l a t i o n ) . 135 4.2.3 L i m i t a t i o n s Of The D i s c r e t e Formulation The n a t u r a l l y continuous s t a t e v a r i a b l e (cu.m./ha.) of a treatment u n i t of Scots pine, as s i m u l a t e d by K i l k k i 1 s model, was adequately represented by d i s c r e t e d e n s i t y l e v e l s . C o m plications a r i s e , however, f o r models with m u l t i d i m e n s i o n a l s t a t e spaces, such as Goulding's D o u g l a s - f i r model. As d e s c r i b e d i n S e c t i o n 4.1.4, the s t a t e of Goulding's model i s the stand age and a l i s t of sample stem diameters at breast height. The stand age i s i n c o r p o r a t e d i n t o the d e f i n i t i o n of the dynamic programming stage, l e a v i n g the dbh d i s t r i b u t i o n to be transformed i n t o an a c c e p t a b l e set of d i s c r e t e s t a t e s . One approach would be to record the p r o p o r t i o n s of the stand basal area i n a number of dbh c l a s s e s d e f i n e d about the average stand d b h 1 1 . For example, the f i v e diameter c l a s s e s t h a t Goulding uses to s p e c i f y crown t h i n n i n g s together with the average stand diameter, might adequately express the s t a t e i n a d i s c r e t e form. However, i f the p r o p o r t i o n of the stand basal area f a l l i n g i n each diameter c l a s s i s expressed i n increments of .1, and the average stand diameter takes on as few as 10 v a l u e s , the r e s u l t i n g s e t of d i s c r e t e s t a t e s i s p r o h i b i t i v e l y l a r g e . Without gross s i m p l i f i c a t i o n of the s t a t e d e f i n i t i o n , a f o r m u l a t i o n of Goulding's model i n a network problem i s c o m p u t a t i o n a l l y i n f e a s i b l e . Consider the f o l l o w i n g compromise approach. The l a r g e 1 1 T h i s r e p r e s e n t a t i o n of the s t a t e of a stand has been used i n a growth and y i e l d model f o r Monterey pine (Pinus r a d i a t a ) , by C l u t t e r and A l l i s o n (1974). 136 number of p o s s i b l e diameter d i s t r i b u t i o n s t a t e s i s r e p l a c e d by s t a t e s that r e c o r d only t h a t a management a c t i o n has o c c u r r e d . For example, s t a t e 4 might be d e f i n e d as the c o n d i t i o n of the stand a f t e r i t has been thinned by 25% cf i t s basal area, f o r the f i r s t time. With the network s o l u t i o n d e s c r i b e d above, the network i s processed i n a forward manner, from l e f t to r i g h t . At each a c t i v i t y s t a t e , the dbh d i s t r i b u t i o n r e s u l t i n g from the optimal pathway to t h a t s t a t e - s t a g e node i s s t o r e d . The dbh d i s t r i b u t i o n i s not used as an i n p u t / o u t p u t s t a t e t o any stage, but only to compute the r e t u r n f o r a t r a n s i t i o n . The i m p l i c a t i o n s of t h i s approach are most e a s i l y d e s c r i b e d with r e f e r e n c e to an example, Figure 19a. Consider the t r a n s i t i o n s cut of s t a t e 4 ( f i r s t t h i n n i n g ) i n t o s t a t e 5 (a second t h i n n i n g ) or s t a t e 6 (a c l e a r c u t ) . The s o l u t i o n a l g o r i t h m i s based on the Markov assumption t h a t the only i n f o r m a t i o n about previous stages r e l e v a n t to s e l e c t i n g optimal values f o r the c u r r e n t d e c i s i o n v a r i a b l e s i s summarized by the s t a t e v a r i a b l e 1 2 . O b v i o u s l y , the i n f o r m a t i o n t h a t the stand has been thinned once by 25% of i t s b a s a l area i s not s u f f i c i e n t knowledge on which to base the d e c i s i o n whether to schedule a second t h i n n i n g or proceed d i r e c t l y to the c l e a r c u t . The optimum pathway to s t a t e 4 might i n c l u d e the low d e n s i t y l e v e l s t a t e 3, because of the a s s o c i a t e d low p l a n t i n g c o s t . Yet i t i s c o n c e i v a b l e t h a t a second t h i n n i n g , s t a t e 5, might be on the f i n a l optimum pathway i f the i n i t i a l d e n s i t y were higher i 2 s e e wagner (1969, p. 343) f o r a c o n c i s e d e s c r i p t i o n of the c o n d i t i o n s and assumptions of m u l t i s t a g e a n a l y s i s . ,137 State Description 1 i n i t i a l (dummy) state 2 1200 stems /acre at age 20 years 3 600 stems /acre at age 20 years 4 thinning state, removed 25% of stand basal area 5 thinning state, removed.-'15% of stand basal area 6 clea r c u t Figure 19. Goulding's model i n a network de c i s i o n problem, with a s i m p l i f i e d d e f i n i t i o n of state (a), and restructured to avoid non-Markov s i t u a t i o n s (b). 138 ( s t a t e 2). The d e f i n i t i o n of s t a t e 4 c a r r i e s too l i t t l e i n f o r m a t i o n about the previous t r a n s i t i o n s to allow the t r a d e o f f between d e n s i t y c o s t s and p r o f i t s from a second t h i n n i n g to be e v a l u a t e d . These s i t u a t i o n s can o f t e n be circumvented through r e s t r u c t u r i n g the problem, although with much l o s s of e f f i c i e n c y i n the s o l u t i o n process. E s s e n t i a l l y , whenever necessary, pathways are separated by adding redundant s t a t e s , so t h a t the Markov assumption holds i m p l i c i t l y . For example, in F i g u r e 19b, the s t a t e s 4' and 5' have been added, i d e n t i c a l i n d e f i n i t i o n to 4 and 5 r e s p e c t i v e l y . However, a d d i t i o n of the two s t a t e s r e s u l t s i n separate pathways corresponding to each of the d e n s i t y l e v e l s (2-4-5'-6) and (2-4'-5-6). The s t a t e 5 1 can be i n t e r p r e t e d as a t h i n n i n g of 15% b a s a l area a f t e r being i n s t a t e 2, 1200 stems per acre at age 20. The p r e v i o u s s t a t e i s i m p l i c i t l y r e c o g n i z e d i n the d e f i n i t i o n of s t a t e 4, and the t r a n s i t i o n s 4-5* or 4-6 can be evaluated u t i l i z i n g the Markov assumptions. On s e l e c t i n g the t r a n s i t i o n i n t o s t a t e 6, the t r a d e o f f between stand d e n s i t y at age 20 years and the second t h i n n i n g i s e v a l u a t e d . E s s e n t i a l l y , r e s t r u c t u r i n g the problem i n t h i s manner f o r c e s the s o l u t i c n program to enumerate and compare management sequences that d i f f e r i n the management a c t i o n s i n c l u d e d . The dynamic programming decomposition advantages are used only to determine the t i m i n g of the management a c t i o n s . fin example i s provided i n Appendix VIII t h a t uses Goulding's model i n the network f o r m u l a t i o n to c o n s i d e r three l e v e l s of s t o c k i n g and up to two t h i n n i n g s . 139 4.2.4 Summary And D i s c u s s i o n The g r e a t e s t disadvantage of the network approach to s o l v i n g HP2, the d e c i s i o n problem of f i n d i n g the o p t i m a l management seguence f o r a treatment u n i t , i s the p r o h i b i t i v e l y l a r g e set of s t a t e s necessary to approximate the c o n t i n u o u s , m u l t i d i m e n s i o n a l s t a t e space of any but the s i m p l e s t stand growth model. The network f o r m u l a t i o n was demonstrated t o be q u i t e e f f e c t i v e with K i l k k i ' s Scots pine model (a s i n g l e s t a t e v a r i a b l e - volume), but r e q u i r e d s i m p l i f i c a t i o n and r e s t r i c t i o n s when a p p l i e d t o Goulding's D o u g l a s - f i r model, where the s t a t e i s , e s s e n t i a l l y , the dbh d i s t r i b u t i o n of the stand. However, the s o l u t i o n procedure i s f a s t and i n e x p e n s i v e , due to the advantages of the dynamic programming decomposition and the e f f i c i e n c y of the computer generated graph s t r u c t u r e of a l t e r n a t i v e management sequences. Apart from problems of s t a t e dimension, stand models that are to be used as t r a n s i t i o n f u n c t i o n s i n multistage f o r m u l a t i o n s should have the c h a r a c t e r i s t i c s recommended i n S e c t i o n 4.1.6 f o r models s u b j e c t e d to d i r e c t o p t i m i z a t i o n . The major advantage of the network f o r m u l a t i o n i s t h a t i t converges to an a c c e p t a b l e p o l i c y without i n t e r v e n t i o n . Conseguently, the network f o r m u l a t i o n i s s u i t a b l e f o r o p t i m i z i n g MP2 as a submodel under c o n t r o l of an automatic process, the decomposition model f c be d e s c r i b e d i n S e c t i o n 5. 140 5 . J o i n t O p t i m i z a t i o n Of «nM find MP2 Via Decomposition In Sec t i o n 3.4.3 i t was e s t a b l i s h e d that a l i n e a r programming model was the most e f f i c i e n t f o r m u l a t i o n of MP1, the commodity a l l o c a t i o n problem. Conversely, a s o l u t i o n t o MP2, the s c h e d u l i n g of management a c t i o n s on a treatment u n i t , should e x p l o i t the m u l t i s t a g e s t r u c t u r e of the problem. A s y n t h e s i s of l i n e a r programming with m u l t i s t a g e decomposition was d e s c r i b e d , using the Lagrange m u l t i p l i e r s generated by the LP master problem t o c o o r d i n a t e the o p t i m i z a t i o n of one or more subproblems. T h i s approach, Dantzig-Wolfe decomposition, was motivated i n S e c t i o n 3.4.4 through an a n a l y s i s of the r o l e of the Lagrange m u l t i p l i e r s i n the d i s c r e t e maximum form of the subproblem. In t h i s chapter, Dantzig-Wolfe decomposition w i l l be used to l i n k the Timber RAM LP model with the network f o r m u l a t i o n of the subproblem. 5.1 The l i n e a r Program Master Problem The l i n e a r approximation to MP 1 a l l o c a t e s c e r t a i n c o n s t r a i n e d commodities to management sequences f o r a l l treatment u n i t s across a management u n i t . For t h i s study. Timber BAM was s e l e c t e d to form the b a s i s of the MP1 l i n e a r model, due to i t s e x t e n s i v e o p e r a t i o n a l use by the U.S. F o r e s t S e r v i c e and BCFS i n t e r e s t i n the system. The Timber RAM software c o n s i s t s of a matrix generator, which c r e a t e s a l i n e a r model i n a form 141 s o l v a b l e by a commercially a v a i l a b l e LP code, and a r e p o r t w r i t e r that t r a n s l a t e s the LP s o l u t i o n i n t o u s e f u l t a b l e s and graphs. The s t r u c t u r e of the l i n e a r model i s well documented {Nazareth, 1S 71) and only those elements r e l e v a n t to the decomposition model w i l l be reviewed here. The n o t a t i o n w i l l c l o s e l y f e l l o w t h a t of S e c t i o n s 1 and 3.4.3. Timber BAM a l l o c a t e s on time increments of 1 decade, and a planning horizon of 10 decades was used throughout the study. Two v a r i a b l e s must be d e f i n e d . Let x be the area of uk treatment u n i t u managed by the a l t e r n a t i v e management seguence (k) a . The t o t a l volume cut i n decade j i s C . u j The values t h a t these two v a r i a b l e s can take on are r e l a t e d and l i m i t e d by a s e t cf c o n s t r a i n t e guations. The area of a treatment u n i t t h a t i s managed i s , of course, l i m i t e d to the t o t a l area of the treatment u n i t : , uk — u The r e l a t i o n s h i p between x and C i s e s t a b l i s h e d through uk j the e q u a l i t y Z E x J2_2 " C. =0 j=1, J (5.1-2) uk -i u k gx , J uk 3C. where — J - i s the volume per u n i t area harvested i n decade j . Volume flow c o n s t r a i n t s l i m i t the r a t e o f change of harvest volumes between time p e r i o d s . The c o n s t r a i n t used throughout 142 t h i s study was t h a t the volume harvested i n p e r i o d j must be w i t h i n ± 10% of the volume harvested i n p e r i o d j-1, i . e . c. < l . i c . , j=1, . .., J (5.1-4) C j ±- °-9Cj-l * = 1 J (5.1-3) (C 0 i s the volume cut i n the present decade.) The IE o b j e c t i v e was to maximize the present net worth of the management u n i t : u K M a x i m i z e R = Z Z x — (5.1-5) u k u k 3x , u k 3 R where — i s the present net worth of a u n i t c f area of 3 x u k treatment u n i t u, when managed by a l t e r n a t i v e a • 3 C U The commodity input-output c o e f f i c i e n t s — 3 — and p r i c e 9 x , a R u k c o e f f i c i e n t are organized as uxK v e c t o r s : o x u k { > 1 / 2 , . . . , J , . . . } 3x 3x . 3x , 3x , u k u k u k u k (5.1-6) A d d i t i o n a l s t r u c t u r a l c o e f f i c i e n t s u s u a l l y i n c r e a s e the dimension c f the v e c t o r to > J+1 elements. T h i s e x p r e s s i o n of the management seguence i s known as the LP vect o r or a c t i v i t y . 143 5.2 Decomposition Cf The l i n e a r Model Each i t e r a t i o n of the LP simplex a l g o r i t h m 1 3 r e s u l t s i n a f e a s i b l e a l l o c a t i o n of the c o n s t r a i n e d commodities. The d u a l v a r i a b l e s at each i t e r a t i o n are the marginal values of the commodities corresponding to the a l l o c a t i o n . In s e c t i o n 3.4.3, the marginal values were d e s c r i b e d as ' c o n s t r a i n e d 1 d e r i v a t i v e s , and i n t e r p r e t e d as the r a t e of change cf the o b j e c t i v e f u n c t i o n r e s u l t i n g from f e a s i b l e p e r t u r b a t i o n s of the commodities. The marginal value of the commodity volume harvested i n decade j i s the dual v a r i a b l e a s s o c i a t e d with Eg. (5.1-2), , <5 C , D j=1,...,J, In the decomposition a l g o r i t h m , these commodity p r i c e s are i n c o r p o r a t e d i n t o the o b j e c t i v e f u n c t i o n of the MP2 subproblem. Eg. (3.3-20). T R = Z R . (v . ,a ) u . ut ut ur The amount cf commodity j produced on treatment u n i t u, at time t , by management a c t i o n a , i s c . The commodity weighted by ut j u t i t s marginal value, i s added to the r e t u r n f u n c t i o n of MP2: T R = £ ( R + Z c , 9R . f 5 2-1\ u t = 1 ut . ]ut (5.-2 D The simplex a l g o r i t h m i s the LP s o l u t i o n process d e v i s e d by George Dantzig and should not be confused with the s e g u e n t i a l simplex search technique d e s c r i b e d i n S e c t i o n 4.1,1. 144 When the management a c t i o n s form a l t e r n a t i v e management sequence k, the r e t u r n per u n i t area of treatment k managed by (k) . a i s u 6 R T 9 R 3 c . 6 Xuk t=l 3 X u t j 8 Xuk 6 C j Note that t h i s form of the MP2 o b j e c t i v e f u n c t i o n i s e q u i v a l e n t to Eg. (3.4-18), and that s o l u t i o n of the subproblem maximizes the d e c i s i o n d e r i v a t i v e of the management a l t e r n a t i v e i n terms of the LP master problem. In e f f e c t , when the subproblem consumes/produces a commodity that i s s c a r c e i n the LP master problem, i t b u y s / s e l l s the commodity from/to the master problem at the marginal value. Consequently, the subproblem i s s o l v e d to maximize the r e t u r n net of commodity va l u e s . The decomposition a l g o r i t h m used i n t h i s study can be b r i e f l y s t a t e d . . 1 : Assume t h a t the IP master problem has been provided with a candidate s e t of management seguences a c o n s t r u c t e d so t h a t i t c o n t a i n s at l e a s t one f e a s i b l e s o l u t i o n a f . 2 : Optimize the l i n e a r model with the c u r r e n t candidate set a. 3 : Obtain the marginal values of the c o n s t r a i n e d commodities, - — (j=1# J ) . j 4 : For each treatment u n i t , s o l v e the network f o r m u l a t i o n of * MP2 using the augmented o b j e c t i v e f u n c t i o n , i . e . , f i n d a u 145 such t h a t Eg. (5.2-2) i s maximized. 5 : Create LP v e c t o r s from a u (u=1, ..., U) and add the new a c t i v i t i e s i n t o the c u r r e n t a. 6 : Optimize the l i n e a r model with the c u r r e n t c a n d i d a t e s e t a. I f t h i s step r e s u l t s i n an improved s o l u t i o n , continue with step 3. 7 : The c u r r e n t LP s o l u t i o n i s o p t i m a l . 5.2.1 Dantzig-Wolfe Decomposition With MPSX The program implementing the decomposition a l g o r i t h m u t i l i z e s the IBM Mathematical Programming System Extended (MPSX) (IBM, 1971a) as we l l as the FORTRAN network s o l u t i o n system SOLVE d e s c r i b e d and demonstrated i n S e c t i o n 4.2 . A fl o w c h a r t (Figure 20) of the decomposition program i s provided and the MPSX program i s l i s t e d i n Appendix IX. The program s t a r t s under c o n t r o l of MPSX. A FORTRAN procedure SUBIN i s invoked which reads i n var i o u s problem parameters and f l a g s . The MPSX READCOMM f a c i l i t y (IBM, 1971b) i s used to t r a n s f e r data between the FORTRAN subsystems and MPSX. On r e t u r n i n g t o MPSX, the LP model (matrix) i s read o f f a l i n e f i l e and loaded onto the problem f i l e . The problem i s then set up i n core and o p t i m i z e d . On convergence to an optimum, the c u r r e n t b a s i s i s saved on the problem f i l e and the marginal values of the c o n s t r a i n e d commodities are s e l e c t e d and loaded i n t o MPSX v a r i a b l e s . C o n t r o l i s then t r a n s f e r r e d to the FORTRAN procedure SUBDC which loads 1 4 6 M P S X START i LP F i l e (matrix) CONVERT Output Procedures FORTRAN SUBIN' Problem op-lions and /parameters. 0 Graph F i l e SOLVE Network Sol-ution prog. Generate LP Vectors * _ user supplied MPSX procedures regular MPSX procedure are underlined Figure 2 0 . Flowchart of the MPSX Dantzig-Wolfe decomposition program. 147 and t r a n s f e r s c o n t r o l to the network s o l u t i o n subsystem SOLVE. Under c o n t r o l of program SOLVE the network models are rea d o f f the GRAPH f i l e , and set up i n core. The marginal values of the commodities are obtained from MPSX v i a READCOMM, and the network subproblems are s o l v e d . The o p t i m a l management seguences, expressed as Timber RAM LP v e c t o r s , are w r i t t e n to a l i n e f i l e i n the format of an MPSX REVISE f i l e . C o n t r o l i s then t r a n s f e r r e d back to MPSX and the SOLVE system i s unloaded. Under c o n t r o l of MPSX, the new LP v e c t o r s i n the REVISE f i l e are added to the model on the problem f i l e . The r e v i s e d problem i s then s et up, the previous optimal b a s i s r e s t o r e d , and op t i m i z e d . I f no improvement r e s u l t s from the o p t i m i z a t i o n , the c u r r e n t s o l u t i o n i s con s i d e r e d o p t i c a l and output procedures are invoked, f o l l o w e d by t e r m i n a t i o n of execu t i o n . I f improvement i n the o b j e c t i v e d i d occur, the new marginal values are s e l e c t e d and c o n t r o l i s again t r a n s f e r r e d to the SOLVE subsystem. 5.3 An Example Problem To Demonstrate The Decomposition Approach In t h i s s e c t i o n , an example problem i s d e s c r i b e d i n v o l v i n g the management of 85,000 hectares of Sects pine. A demonstration of the decomposition approach with Goulding's D o u g l a s - f i r model would be more a p p r o p r i a t e f o r a t h e s i s w r i t t e n i n B r i t i s h Columbia. However, K i l k k i ' s Sects pine model i s cheaper to optimize and i n c l u d e s a more complete economic model. Otherwise, the example problem i s g u i t e r e a l i s t i c i n s i z e and complexity. 148 5.3.1 Problem D e s c r i p t i o n The management u n i t c o n s i s t s of e i g h t treatment u n i t s of Scots pine. Each treatment u n i t i s d e s c r i b e d below and the management o u t l i n e r e p o r t i s co n t a i n e d i n Appendix X. The growth curves are those represented i n F i g u r e 10. Treatment Unit T01 Treatment u n i t 1 (T01) i s 15,000 hecta r e s of overmature timber, with an average of 270 cu.m./ha. The estimated value of the timber, net of h a r v e s t i n g c o s t s , i s 5 $/cu.m. The timber i s to be harvested i n the next 20 years. Three r e g e n e r a t i o n options are to be c o n s i d e r e d . High d e n s i t y p l a n t i n g at a cost of 30 $/ha. w i l l place the stands cn growth curve 9, while p l a n t i n g to a lower d e n s i t y (curve 8) r e s u l t s i n a lower c o s t , 24 $/ha. Nat u r a l r e g e n e r a t i o n (curve 7) r e g u i r e s s i t e p r e p a r a t i o n c o s t i n g $10 $/ha. Depending on the type of r e g e n e r a t i o n , up t o 3 t h i n n i n g cuts may be scheduled. T02 Treatment u n i t 2 (T02) i s 5,000 h e c t a r e s of timber s i m i l a r to T01, except t h a t i t i s more remote. Consequently, the t r a n s p o r t a t i o n c o s t component of any management a c t i o n i s higher. The u n i t w i l l y i e l d 310 cu.m./ha. with a value of 5 $/cu.m. The r e g e n e r a t i o n o p t i o n s are i d e n t i c a l to those 149 of T01; high, d e n s i t y p l a n t i n g , low d e n s i t y p l a n t i n g , and n a t u r a l r e g e n e r a i c n . However, t r a n s p o r t a t i o n f a c t o r s i n c r e a s e p l a n t i n g c o s t s to 35 and 29 $/ha, r e s p e c t i v e l y , while s i t e p r e p a r a t i o n c o s t s are unchanged (10 $/ha,). S i m i l a r l y , t h i n n i n g and f i n a l harvest c o s t s are in c r e a s e d by 3 $/cu.m. , and 2 $/cu.m., r e s p e c t i v e l y . T03 Treatment u n i t 3 (TG3), c o n s i s t i n g of 10,000 h e c t a r e s , i s i d e n t i c a l i n management o p t i o n s , c o s t s and r e t u r n s t o T01, but i s c u r r e n t l y i n a c c e s s i b l e . Consequently, the e a r l i e s t harvest cut cannot be scheduled u n t i l the second decade of the planning p e r i o d , T04 Treatment u n i t 4 (T04) i s timber of age 20 ye a r s , t h a t was plan t e d at high d e n s i t y (curve 9). There are 20,000 hectares of T04. Up to three t h i n n i n g c u t s may be scheduled before the harvest c u t , T05 Treatment u n i t 5 (T05) i s the same high d e n s i t y as T04, but i s c u r r e n t l y 35 years o l d . Up to three t h i n n i n g c u t s may be scheduled before the harvest cut on the 15,000 hectares of T04. T06 Treatment u n i t 6 (T06) i s c u r r e n t l y 50 years o l d and of medium d e n s i t y (curve 8). The 10,000 hectares may be thinned up to three times before the f i n a l h a r v e s t cut. T07 Treatment u n i t 7 (107), with an area of 10,000 h e c t a r e s , i s i d e n t i c a l t o TC6 except t h a t h a r v e s t i n g c o s t s are 2 $/cu.m. higher. T08 Treatment u n i t 8 (T08) i s i d e n t i c a l to T07 except t h a t i t i s 10 years o l d e r (60 y e a r s ) . The area of T08 i s 10,000 hecta r e s . 150 The time frame o f the planning problem was d e f i n e d t o decrease i n r e s o l u t i o n towards the planning h o r i z o n . The f i r s t 10 stages each have a d u r a t i o n of 1 year, the next 10 stages are 2 y e a r s , and the remaining stages are each a decade i n l e n g t h . Consequently, those treatment u n i t s t h a t have management a c t i o n s e a r l y i n the planning p e r i o d have many more a r c s and pathways i n t h e i r network f o r m u l a t i o n s than those treatment u n i t s with management a c t i o n s l a t e r i n the pl a n n i n g p e r i o d . For example, the management o u t l i n e r e p o r t (Appendix X) f o r T02, which has three r e g e n e r a t i o n o p t i o n s over the f i r s t 20 ye a r s , computes 8695 f e a s i b l e management sequences i n i t s network f o r m u l a t i o n . Treatment u n i t 4, however, does not have any management o p p o r t u n i t i e s u n t i l the f o u r t h decade, and i t s network model c o n t a i n s only 316 a l t e r n a t i v e management seguences. In t o t a l , the network models f o r a l l the treatment u n i t s r e p r e s e n t 28,341 a l t e r n a t i v e management seguences. 5.3.2 The Unconstrained Optimal S o l u t i o n Each network model was optimized by the SOLVE programs. The op t i m a l p o l i c i e s are represented i n Figure 21 and Figure 22, as the volume per acre on each treatment u n i t over the planning p e r i o d . The s o l i d l i n e r e p r e s e n t s the op t i m a l management seguence found by the network problem, and the dashed l i n e i s the p r o j e c t i o n c f the management sequence to the end of the planninq p e r i o d . (The network model assumes t h a t the optimal T03: overmature, in a c c e s i b l e u n t i l second decade 1 1 1 1 1 1 1—™n 1 1 1 1 1 2 3 ^ 5 6 7 8 9 10 11 12 D e c a d e s TO4: age 20, density curve 9 T I I I I ^ I i I I I I I 1 2 3 4 5 6 7 8 9 1 0 11 12 D e c a d e s 21. Unconstrained optimal management sequences for treatment units 1-4 T05: age 35 years, high density (curve 9) / i , / i / / i ' / i "T I 1 1 1 "T 1 1 1 2 3 4 5 6 7 8 9 10 11 12 41 T06: age 50 years, medium density (curve 89 n / 1 *• , "I ~ l 1 r~ "| 1 r 1 2 3 /, 5 6 7 8 9 1 0 i r 12 CD i _ U 3 24 [A TO7: age 50 years, curve 8, high harvest costs (+ $2) ~i i i r i 1 1 f 1 1 1 1 2 3 U 5 6 7 8 9 10 11 12 D e c a d e s 3 2 1 1 T08: age 60 years, curve 8, high harvest costs (+ $2) A ~1 I 1 T 1 1 j -1 2 3 h 5 6 7 8 9 10 11 12 Decades Figure 22. Unconstrained optimal management sequences f or treatment units 5-8. 153 management seguence w i l l be repeated i n f i n i t e l y ) . Treatment Unit T01 Treatment u n i t 1 i s scheduled to l i q u i d a t e i t s s t a n d i n g timber immediately and plant to the high d e n s i t y . Thinning cuts of medium i n t e n s i t y (60 < cu.m./ha. < 90) are to be made at 45 years and 5 5 years, f o l l o w e d by a f i n a l harvest cut at 65 years. TO2 Treatment u n i t 2, with a g r e a t e r standing volume and higher h a r v e s t i n g c o s t s , i s s i m i l a r l y scheduled to l i g u i d a t e i t s standing timber and regenerate to a high d e n s i t y . However, two heavy t h i n n i n g s ( >90 cu.m./ha.) are scheduled at 45 years and 65 years, with the f i n a l h a r vest cut during decade 7. T03 As soon as treatment u n i t 3 i s a c c e s s i b l e (at the end of the f i r s t decade), the s t a n d i n g timber i s c u t . The u n i t i s p l a n t e d at the high d e n s i t y . A medium i n t e n s i t y t h i n n i n g i s scheduled at 55 years with a heavy t h i n n i n g a decade l a t e r . The harvest cut occurs a t 85 years. T04 Treatment u n i t 4 grows undisturbed u n t i l 30 years i n t o the p l a n n i n g p e r i o d . A medium t h i n n i n g , a l i g h t t h i n n i n g ( <60 cu,m./ha.), and a medium t h i n n i n g are p r o j e c t e d f o r 30 years, 38 years and 45 years, r e s p e c t i v e l y . The c l e a r c u t occurs at 55 years, T05 Treatment u n i t 5 i s to be thinned (medium i n t e n s i t y ) at 15 years, followed by a heavy t h i n n i n g at 21 yea r s , and a f i n a l h a rvest cut at 35 years. 154 T06 Treatment u n i t 6 i s scheduled f o r an immediate l i g h t t h i n n i n g c u t , followed by a heavy t h i n n i n g at 9 years. The f i n a l harvest cut occurs a t 21 years. T07 The optimal management seguence f o r T07 i s i d e n t i c a l t o t h a t of T06, except t h a t the harvest cut i s delayed u n t i l year 25. T08 An immediate heavy t h i n n i n g cut i s scheduled f o r treatment u n i t 8, with a harvest cut a t 15 years. The t o t a l volume scheduled f o r h a r v e s t i n g i n each decade i s presented i n F i g u r e 23a. Large f l u c t u a t i o n s occur i n the volume flow o f f the u n i t , e s p e c i a l l y i n the f i r s t 2 decades of the plann i n g p e r i o d when the c u r r e n t s t a n d i n g timber i s being l i g u i d a t e d . These f l u c t u a t i o n s would probably make the plan unacceptable. However, the present net worth of the unconstrained p l a n , $89.8 x10&, p r o v i d e s an upper l i m i t with which one can compute the o p p o r t u n i t y c o s t s of management plans i n v o l v i n g c o n s t r a i n e d volume f l e w s . 1 5 5 14- Q unconstrained optima for each 12- \ treatment unit 10-8- \ <a> 6-4-2-1 — i 1 1 1 1 1 — i 1 — i — 1 2 3 4 5 6 7 8 9 10 o x u a> +-l/t > o X E O > K H 8 6 4-2 10 8H 6 4 2 f irst feasible solut ion _ scaled marginal values of volume per acre 1 2 3 4 5 6 7 8 9 10 f i n a l , " o p t i m a l solution / I + ' O — ° -CO \ 40 [-36 32 h28 24 20 0) C £<o « ° a> «/»• Z — (b) (c) — i 1 1 1 1 1 1 1 1 r 1 2 3 4 5 6 7 8 9 10 Decades F i g u r e 2 3 . V o l u m e f l o w g r a p h s f o r t h r e e m a n a g e m e n t u n i t p l a n s . 156 5.3.3 S t a r t - u p Procedure And F i r s t F e a s i b l e S o l u t i o n A l i n e a r model of the management u n i t was created with the Timber RAM matrix generator {Navon, 1971b), The LP o b j e c t i v e was to maximize the present net worth over a planning p e r i o d of 10 decades. A volume flow c o n s t r a i n t was imposed t o l i m i t the volume cut each decade to wi t h i n + 10% of the previous decade cut . The c u r r e n t decade cut was set at 5x10 6 cu.m./ha. Only the s t r u c t u r a l elements o f the matrix were generated; the Timber RAM LP a c t i v i t i e s were omitted. The LP v e c t o r s corresponding to the unconstrained optimal management sequence were a u t o m a t i c a l l y produced i n MPSX format by the network SOLVE system. These v e c t o r s were s u p p l i e d to the matrix together with a set of a r t i f i c i a l v e c t o r s that s a t i s f i e d the problem c o n s t r a i n t s but c o n t r i b u t e d l a r g e negative p e n a l t i e s to the o b j e c t i v e f u n c t i o n . The MPSX program was executed f o r three i t e r a t i o n s of the decomposition procedure. A f t e r three decompositions, a l l the a r t i f i c i a l v a r i a b l e s had been e l i m i n a t e d from the s o l u t i o n . F i g u r e 23b shows the smooth trend i n volume cut per decade over the planning p e r i o d . The marginal values of volume produced each decade are repre s e n t e d on the graph by arrows; the d i r e c t i o n and l e n g t h of the arrow r e p r e s e n t i n g , r e s p e c t i v e l y , the s i g n and magnitude of the marginal value. Hence, from the graph one can see th a t there i s a r e l a t i v e l y l a r g e penalty a g a i n s t volume p r o d u c t i o n i n decade 1, due t o the e x c e s s i v e volumes of standing timber on T01 and T02 that must be l i q u i d a t e d before year 20 of the planning 157 p e r i o d . P o s i t i v e marginal values at decades 3, € and 7 i n d i c a t e a shortage of volume i n these decades. The marginal value arrows can be regarded as f o r c e s modifying the shape of the volume flow curve. At the next decomposition, the marginal values w i l l weight the network s o l u t i o n process to f a v o r management seguences that change the curve i n the i n d i c a t e d manner. 5,3,4 F i n a l S o l u t i o n The system was r e s t a r t e d and executed f o r s i x more decompositions, f o r a t o t a l of nine decomposition i t e r a t i o n s . The flow of volume and net revenue o f f the management u n i t are d i s p l a y e d i n F i g u r e 23c. Note that net revenue, as a commodity, cou l d have been c c n s t r a i n e d with a flow p o l i c y analogous t o the volume c o n s t r a i n t s . The decomposition a l g o r i t h m s would then s o l v e the network subproblems weighted by the marginal values of both volume and net revenue. A f t e r nine decompositions, the problem had net converged, i . e . , the n i n t h decomposition had produced p r o f i t a b l e management seguences which had been i n c o r p o r a t e d i n t o the LP b a s i s . The d e c i s i o n t o stop at nine decompositions was made through examining a p l o t of the LP o b j e c t i v e f u n c t i o n value a g a i n s t the LP i t e r a t i o n s . (Figure 24). A f t e r seven decompositions, the o b j e c t i v e value i n c r e a s e s only s l i g h t l y . Although i t i s p o s s i b l e that the procedure simply 1 5 8 Iterations Figure 24. Stopping r u l e - asymtotic behavior of the decomposition model objective function. 159 found a p l a t e a u on the o b j e c t i v e s u r f a c e and could undergo f u r t h e r s u b s t a n t i a l improvement, i t i s more l i k e l y t h a t the model i s very c l o s e to i t s o p t i m a l p o l i c y . The unconstrained optimal r e t u r n c f $89.fi x10 6 provides an upper bound on the c o n s t r a i n e d optimal r e t u r n . Proximity of the o b j e c t i v e value to i t s upper bound i n c r e a s e s c o n f i d e n c e i n t h i s h e u r i s t i c s topping r u l e . , On examining the p o l i c i e s s e l e c t e d f o r the i n d i v i d u a l treatment u n i t s , i t was found t h a t 322 hectares o f T02 and a l l 10,000 hectares of T03 are l e f t unmanaged. Because of the even flow c o n s t r a i n t of ± 10% f l u c t u a t i o n i n volume cut per decade, these treatment u n i t s cannot be i n c l u d e d i n the management u n i t . As they are now d e f i n e d , T02 and T03 must l i q u i d a t e t h e i r s t a n d i n q timber w i t h i n the f i r s t two decades of the planning p e r i o d , when T01 has already produced a s u r f e i t of volume. A s o l u t i o n might be to allow the timber cn these u n i t s to be h e l d u n t i l the t h i r d or f o u r t h decade before c u t t i n g , or to r e l a x the even flew p o l i c y i n the f i r s t few decades. Note that the o p p o r t u n i t y cost of the even flow c o n s t r a i n t i s s i z e a b l e (89.8 -74,2 = $15.6 x10*) , Although most of the cost i s due to the e x c l u s i o n of areas of T02 and T03, a s i g n i f i c a n t c o s t can be a t t r i b u t e d t o the n e c e s s i t y of s a t i s f y i n g the volume flow c o n s t r a i n t s with i n e f f i c i e n t management seguences. Even i f the excluded areas were managed with t h e i r i n d i v i d u a l optimum management seguences, the r e t u r n ($96 x10 6) would l e a v e (89.8 (74.2 + 9.6)=) $6 x10* as the o p p o r t u n i t y c o s t . The management sequences f o r each treatment u n i t t h at were s e l e c t e d as optimal f o r the manaqement u n i t p l a n , are 160 repr e s e n t e d i n terms of volume i n F i g u r e s 25 and 26. The a l t e r n a t i v e management sequences on the f i g u r e s are l a b e l l e d with the areas that they manage. Treatment Unit T01 Treatment u n i t 1 has two a l t e r n a t i v e management seguences. The o r i g i n a l unconstrained o p t i m a l management seguence i s a p p l i e d to 1750 h e c t a r e s . Most of T01 (13,250 hectares) i s managed by a seguence t h a t delays the t h i n n i n g and subseguent harvest c u t s by one decade, r e s u l t i n g i n g r e a t e r volume y i e l d s . T02 Treatment u n i t 2 has two a l t e r n a t i v e management sequences, n e i t h e r of which i s the optimal unconstrained v e r s i o n . Both management sequences delay the f i r s t t h i n n i n g by one decade but the subsequent harvest cut i s unchanged from the unconstrained optimum, at 75 years. T04 Treatment u n i t 4 has the most complicated set of a l t e r n a t i v e management seguences. In g e n e r a l , the f i r s t t h i n n i n g a c t i v i t y , which occurred at 30 years i n the unconstrained optimum seguence, has been delayed u n t i l 35 years, and the three l i g h t t h i n n i n g s have been reduced to two medium t o heavy t h i n n i n g s . One of the three a l t e r n a t i v e management seguences d e l a y s the f i n a l harvest cut to 65 years from 55 years. T05 Treatment u n i t 5 i s managed by a simple management seguence. I t d i f f e r s from the unconstrained optimal seguence i n both the t i m i n g and i n t e n s i t y of t h i n n i n g I* 3 2 to r ZD U C D u / C D 4 ° 3 C D E 2 TOl ,i 13270 ~i i i i 1 J—^—i—i 1 1 1 1— 1 2 3 U 5 6 7 8 9 10 11 12 TO 4 A / / | 11000 / / / / / J / | ]/ / . 1 1 5250 3750-* 1 1 2 3 U 5 6 7 8 9 10 11 12 "i I i i I I I — i 1 1 r 1 2 3 U 5 6 7 8 9 10 11 12 15000 Figure D e c a d es 25. Constrained optimal management sequences f or treatment units 1,2,4, and 5 i i i ( i i i i 1 1 1 1— 1 2 3 U 5 6 7 8 9 10 11 12 D e c a d e s CN1 O Z 2 0; 5750 TO 6 4250 -i i — i 1 1 1 1 1 1 1 1 1— 1 2 3 U 5 6 7 8 9 10 11 12 CD o> 3 £ 2 D O > 1 T08 4900 H I 5100 4 3 2 H TO 7 2050 7950 i i r - 1 i i i 1 1 1 1 1 1— 1 2 3 k 5 6 7 8 9 10 11 12 D e c a d e s Figure i i i i 1 1 1 1 1 1 1 1 2 3 U 5 6 7 8 9 10 11 12 D e c a d es 26. Constrained optimal management sequences f or treatment units 6, 7 and 8. 163 cuts as w e l l as d e l a y i n g the f i n a l harvest cut by one decade. T06 Treatment u n i t 6 has two a l t e r n a t i v e management seguences. Eoth delay the f i r s t t h i n n i n g one decade past t h a t of the unconstrained optimal. In both management sequences, the t h i n n i n g cuts and f i n a l harvest cut are q u i t e c l e a r l y spaced. T07 The two a l t e r n a t i v e management sequences of T07 both delay the i n i t i a l t h i n n i n q , a l l o w i n q heavier t h i n n i n q c u t s . The sequences d i f f e r by one decade i n the ti m i n q o f the f i n a l h a rvest cut. T08 As with T06 and T07, the a l t e r n a t i v e manaqement sequences f o r treatment u n i t 8 delay the i n i t i a l t h i n n i n q by one decade, and provide the o p p o r t u n i t y to delay the f i n a l h arvest c u t . In g e n e r a l , the s e t s of a l t e r n a t i v e management sequences t h a t are optimal within the manaqement context o f the whole management u n i t d i f f e r s u b s t a n t i a l l y from the unconstrained o p t i m a l management seguences f o r each treatment u n i t . 164 5,4 A Goal Parametric A n a l y s i s With The Decomposition Model As s t a t e d i n Se c t i o n 4,1,3, knowledge of the o p t i m a l p o l i c y i s u s u a l l y i n s u f f i c i e n t i n f o r m a t i o n f o r the d e c i s i o n maker. Some knowledge of the behavior of the systems i n the neighborhood of the optimum i s u s u a l l y r e g u i r e d . For example, i n the problem d e s c r i b e d above, one might wish t o ex p l o r e the t r a d e - o f f s necessary to ensure that the cut does not f a l l below 5 x10 6 cu.m. each decade. One approach might be to simply add t h i s c o n s t r a i n t to the l i n e a r model and r e - o p t i m i z e . However, the technigue d e s c r i b e d below, goal parametric a n a l y s i s , provides more i n s i g h t i n t o the dynamics c f problems, by e x p l o r i n g the p o l i c y space between the present optimum and the newly c c n s t r a i n e d optimum. Goal programming was i n t r o d u c e d to the f o r e s t r y l i t e r a t u r e i n response to the d i f f i c u l t y c f ex p r e s s i n g management o b j e c t i v e s i n terms of maximizing cr minimizing a s i n g l e c r i t e r i o n , such as present net worth ( F i e l d , 1 9 7 3 ) 1 4 . With goal programming, the o b j e c t i v e f u n c t i o n i s a weighted sum of va r i o u s c r i t e r i a . For the problem at hand, an i n t e r e s t i n g g o a l o b j e c t i v e would be to maximize the net present worth and, si m u l t a n e o u s l y , minimize the weighted negative d e v i a t i o n s of the decadal volume from the go a l of 5 x10* cu.m. A few s t r u c t u r a l changes must be made t o the l i n e a r model. The minimum cut c o n s t r a i n t d e s c r i b e d above i s 1 4 See Lee (1972) f o r a pragmatic i n t r o d u c t i o n to g o a l programming. B e l l (1975) and Dress (1975) presented t y p i c a l a p p l i c a t i o n s of goal programming i n f o r e s t lands management. 165 C. > 5 xlO j = 1, 2 , (5.4-1) Formulating t h i s c o n s t r a i n t as a goal i s accomplished by e x p l i c i t l y d e f i n i n g the v a r i a b l e D.. as the negative d e v i a t i o n of the commodity from i t s d e s i r e d value, c. + D. ^ 5 x i o 6 j = 1, 2, J (5.4-2) and i n c o r p o r a t i n g D. i n t o the o b j e c t i v e f u n c t i o n . U K 8R J Maximize { I E x , u - W E D . } (5.4-3) u k ^ j 3 uk W i s a parameter t h a t governs the r e l a t i v e importance of the goal i n the o b j e c t i v e f u n c t i o n 1 5 . Note t h a t when H=0, Eg. (5.4-3) i s the o b j e c t i v e f u n c t i o n of the o r i g i n a l l i n e a r model, Eg. (5.1-5) . The procedure f o l l o w e d f o r t h i s demonstration was to i n c r e a s e the parameter 8 over the range [ 0 , P ] , where P was the p o s i t i v e r e a l number above which no change could occur i n the IP s o l u t i o n . The parametric programming f a c i l i t i e s of HPSX were used to maintain an o p t i m a l s o l u t i o n . At f i x e d i n t e r v a l s of W l s True goal programming allows f o r o r d i n a l s o l u t i o n s , i . e . g o als are ranked r a t h e r than weighted, and each g o a l i s optimized i n order. The f o r m u l a t i o n used in t h i s study permits t r a d e - o f f s between the two goals of maximizing PN H and minimizing the s h o r t f a l l from the decadal volume g o a l . 166 (W=.5) the procedure was i n t e r r u p t e d and a decomposition was performed to supply new advantageous management seguences. When the parameter reached a value of W=.97, the MPSX system a s c e r t a i n e d t h a t no f u r t h e r change i n the s o l u t i o n would cccur f o r any l a r g e r w, R e s u l t s are summarized on F i g u r e 27. The graph shows that the goal i s completely s a t i s f i e d when W=,97, but u n s a t i s f i e d i n decades f i v e to e i g h t , when W=.5 . The i m p l i c a t i o n i s that the volume can be in c r e a s e d i n decades nine and ten a t l e s s cost to the o b j e c t i v e of maximizing PNW than i n decades f i v e t o e i g h t . The present net worth of the management u n i t under the new p o l i c y was $72.1 x10*, so the c o s t of a t t a i n i n g the minimum decadal volume goal was $2.1 x10 6. Twenty-six per cent c f the area of the management u n i t was to be subj e c t e d to the management seguences c o n s t r u c t e d d u r i n g the s i n g l e decomposition of the goa l parametric a n a l y s i s . 5.5 Summary And D i s c u s s i o n The Dantzig-Wclfe decomposition model l i n k s the MP 1 and MP2 problems and u t i l i z e s the most e f f i c i e n t f o r m u l a t i o n s f o r each; the a l l o c a t i o n of commodities (MP1) i s solved as an LP problem while the optimum managment seguence (MP2) i s found through a network f o r m u l a t i o n t h a t e x p l o i t s the mu l t i s t a g e nature o f MP2. A c o n v e n t i o n a l l i n e a r programming problem e g u i v a l e n t c f the Dantzig-Wclfe decomposition model d e s c r i b e d above would c o n t a i n 28,341 a l t e r n a t i v e management seguence v e c t o r s , A measure of the O b j e c t i v e : m a x i m i z e PNW + w ( - Zdv ) o w= 0 A w =..S D e c a d e s Figure 27. Decadal volume harvest computed by goal parametric programmi 168 e f f i c i e n c y of the decomposition process i n ge n e r a t i n g u s e f u l IP v e c t o r s i s the area of the management u n i t assigned t o the v e c t o r s r e s u l t i n g from the most re c e n t decomposition. The histograms i n Figure 28 represent the area of the u n i t managed by v e c t o r s c r e a t e d at the i n d i c a t e d decomposition. I t i s apparent from Figure 28 t h a t most of the u n i t i s managed by v e c t o r s produced at the most recent decompositions, i n d i c a t i n g the u t i l i t y of the pro c e s s . fi more r e l e v a n t measure of the e f f i c i e n c y of the decomposition model can be made through examining a breakdown of computer c o s t s f o r the demonstration problem (Table 6 ) . Table 6. Computer c o s t s f o r a demonstration problem of D a n t z i g - H c l f e decomposition. Cost UBC Bate Commercial Percentage Component co s t ($) co s t ( $ ) 1 of t o t a l c o s t V i r t u a l memory 26.90 67.00 32% Pr o c e s s i n g 51.00 127.00 62% SOLVE system 70.00 175.00 -85% T o t a l c o s t 82.50 206.00 * Commercial r a t e s on 0BC«s IBM 370-168 are approximately 2.5 times the u n i v e r s i t y r a t e . Of the t o t a l c o s t of o p t i m i z a t i o n , 851 i s a t t r i b u t a b l e to the SOLVE subsystem which computes the o p t i m a l management seguences f o r each treatment u n i t , at each decomposition. T h i s f i r s t f e a s i b l e s o l u t i o n , 3_| 3 d e c o m p o s i t ions 7 d e c o m p o s i t i o n s o f i n a l so lu t ion , a f t e r 9 deco m posi t ions to cu rd ( _ > a> I goal p a r a m e t r i c s , 1 de compo s i t i o n (#10 ) 3 2 3 2 2i 0 1 2 3 4 5 6 7 8 9 10 i — i 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 F i g u r e 2 8 . 0 1 2 3 4 5 6 7 8 9 10 D e c o m p o s i t i o n I t e r a t i o n s A r e a (%) o f m a n a g e m e n t u n i t s u b j e c t t o t h e m a n a g e m e n t s e q u e n c e s c r e a t e d a t e a c h d e c o m p o s i t i o n . 170 s e r i e s of programs i s h i g h l y g e n e r a l i z e d , i n c o r p o r a t i n g a l l the su b r o u t i n e s of the three stand growth models u t i l i z e d i n the study. The SOLVE system i n c u r s f u r t h e r overhead c o s t s due to the en t r y p o i n t s and in p u t - o u t p u t o p t i o n s that enable i t to be used as a stand alone system, independent of the MPSX master problem. The high v i r t u a l memory component (32% of t o t a l costs) could probably be reduced with improved v i r t u a l memory management i n the SOLVE system. Large a r r a y s are r e q u i r e d t o s t o r e the network problems and i n t e r m e d i a t e r e s u l t s i n the s o l u t i o n process. Heprogramming f o r o p e r a t i o n a l use could s u b s t a n t i a l l y reduce the p r o c e s s i n g c o s t s a s s o c i a t e d with t h i s part of the system. Even with the re c o g n i z e d inadeguacies d e s c r i b e d above, the system compares very f a v o r a b l y with the only a l t e r n a t i v e s o l u t i o n approach - an LP model encompassing a l l the a l t e r n a t i v e management seguences as v e c t o r s . Even i f a matrix o f 28341 v e c t o r s could be c r e a t e d and loaded i n t o a computer, i t seems u n l i k e l y t h a t the c o s t s of o p t i m i z a t i o n would be as l i t t l e as $82.50. 171 6 * C o n c l u s i o n s Planning i s the highest l e v e l of c o n t r o l i n any o r g a n i z a t i o n , and i t i s p r i m a r i l y through p l a n n i n g t h a t a high l e v e l manager e x e r t s i n f l u e n c e over h i s o r g a n i z a t i o n . The planning process i s o f t e n i n f o r m a l : data are c o l l e c t e d , compressed and models a b s t r a c t e d , but the 'model' i s the manager's con c e p t u a l model of the d e c i s i o n problem. In f a c t , even i n o r g a n i z a t i o n s where the process i s f o r m a l i z e d , d e c i s i o n s are r a r e l y made d i r e c t l y cn the r e s u l t s from a model. I n s t e a d , a n a l y s i s of the formal model prov i d e s i n s i g h t s i n t o the d e c i s i o n problem that improves the manager's c o n c e p t u a l model, S technology o f d e c i s i o n a n a l y s i s has a r i s e n to a i d the manager i n o b t a i n i n g these i n s i g h t s from formal models. In the area of f o r e s t lands p l a n n i n g , models e x i s t to s i m ulate the b i o l o g i c a l and economic components of the d e c i s i o n problem. Modern d e c i s i o n a n a l y s i s i s sometimes performed at the management u n i t l e v e l , and l e s s o f t e n at the treatment u n i t l e v e l . T h i s t h e s i s c o n c e n t r a t e s on these two l e v e l s of the p l a n n i n g process, s u b j e c t i n g the d e c i s i o n problem at each l e v e l t o a d e t a i l e d systems a n a l y s i s , with the o b j e c t i v e of e l u c i d a t i n g u n d e r l y i n g mathematical s t r u c t u r e . fit the management u n i t l e v e l , the d e c i s i o n problem was i d e n t i f i e d as the problem of a l l o c a t i n g c o n s t r a i n e d commodities ( l a n d , a l l o w a b l e c u t , c a p i t a l , etc.) to a l t e r n a t i v e seguences of management a c t i o n s . Examination of the problem's c h a r a c t e r i s t i c s showed that the l i n e a r models ccmmonly used at the management u n i t l e v e l are acceptable and pragmatic f o r m u l a t i o n s . D e c i s i o n 172 a n a l y s i s technology i s roost complete at t h i s l e v e l , p r i m a r i l y u t i l i z i n g l i n e a r programming (Timber RAM, MAX-MILLION ). At the treatment u n i t l e v e l , the d e c i s i o n problem i s to schedule management a c t i o n s over the planning p e r i o d i n an optimal manner. T h i s d e c i s i o n problem has a n a t u r a l m u l t i s t a g e s t r u c t u r e t h at should be e x p l o i t e d i n the s o l u t i o n process. The d i f f i c u l t i e s of i n c o r p o r a t i n g complicated stand growth and y i e l d models i n t o the d e c i s i o n model l e d to a c o n s i d e r a t i o n of the g e n e r a l problem of computing optimal management seguences. Two approaches were e v a l u a t e d . A d i r e c t c l i m b i n g a l g o r i t h m , the s e g u e n t i a l simplex search, was implemented i n the framework of a c o n v e r s a t i o n a l s u p e r v i s o r system. The system al l o w s (in f a c t , r e q u i r e s ) a h i g h l y i n t e r a c t i v e d e c i s i o n a n a l y s i s of stand s i m u l a t i o n models, p r o v i d i n g a 'gaming 1 environment g e n e r a l l y c o n s i d e r e d i d e a l f o r d e v e l o p i n g i n s i g h t i n t o management s i t u a t i o n s . Test cases analysed with t h i s approach demonstrate t h a t stand growth models should be c o n s t r u c t e d with d e c i s i o n a n a l y s i s technigues i n mind. T r a d e - o f f s between the v a l i d i t y of the s i m u l a t i o n and a n a l y t i c t r a c t a b i l i t y must be c o n s i d e r e d . A d e s i r a b l e f e a t u r e of t h i s a l g o r i t h m i s t h a t i t s t a r t s with the user's best guess of a d e s i r a b l e seguence of management a c t i o n s and attempts to improve i t . Presumably an experienced manager w i l l provide a s t a r t i n g s o l u t i o n at or near the optimum, reducing search e f f o r t . The second approach embeds the s i m u l a t i o n model i n a network f o r m u l a t i o n , where each pathway through the network c o n s t i t u t e s a f e a s i b l e management seguence. Computer programs generate the network a u t o m a t i c a l l y from a b r i e f o u t l i n e of the 173 management a l t e r n a t i v e s , and s o l v e the network problem i n a separate step. U n l i k e d i r e c t c l i m b i n g , the system runs without i n t e r v e n t i o n , and i s l e s s conducive to post optimal a n a l y s i s . On the other hand, i t i s l e s s l i k e l y t o s t a l l cut on some anomaly of the o b j e c t i v e f u n c t i o n s u r f a c e , f a r from the t r u e optimum. Planning d e c i s i o n s at the management u n i t l e v e l cannot be made independent c f d e c i s i o n s at the treatment u n i t l e v e l . The r e l a t i o n s h i p between the two l e v e l s of the planning process i s e f f i c i e n t l y handled through Wolfe-Dantzig decomposition. A l i n e a r program master problem handles the l i n e a r a l l o c a t i o n problem at the management u n i t l e v e l , but i s l i n k e d to a s e t of subproblems that model the treatment u n i t d e c i s i o n process. The network f o r m u l a t i o n of the subproblem i s i n c o r p o r a t e d i n t o the decomposition system as i t can be executed a u t o m a t i c a l l y from the LP system, and r e q u i r e s no i n t e r v e n t i o n . The decomposition system combines three powerful components. E x i s t i n q p lanning systems t h a t are h i g h l y developed and o p e r a t i o n a l , such as Timber RAM, provide the management u n i t model. The l i n e a r programming i s performed by commercially a v a i l a b l e , s t a t e of the a r t mathematical programming systems, such as IBM MPSX. Fo r e s t stand models provide accurate and d e t a i l e d e stimates of responses t c management a c t i o n s . Furthermore, the decomposition approach i s economicaaly a t t r a c t i v e , s o l v i n g problems of f a r gre a t e r complexity than could be contemplated under c o n v e n t i o n a l LP f o r m u l a t i o n s . The d e c i s i o n a n a l y s i s technigues d e s c r i b e d i n t h i s t h e s i s are a p p l i c a b l e to f o r e s t lands p l a n n i n g i n B r i t i s h Columbia. Both the c o n v e r s a t i o n a l s u p e r v i s o r and network programs should 174 be used to evaluate i n t e n s i v e management p o l i c i e s f o r the more v a l u a b l e B.C. treatment u n i t s , such as high s i t e c o a s t a l D o u g l a s - f i r . The b i o l o g i c a l components of the necessary models are a l r e a d y w e l l developed. The economic components are l a c k i n g : e s t i m a t e s of management a c t i o n c o s t s can be made with some c o n f i d e n c e , but systems to estimate value must be improved. When an a c c e p t a b l e treatment u n i t model has been obt a i n e d , a program of i n t e n s i v e management f o r a c o a s t a l PSYU or TFI cou l d be c o o r d i n a t e d with ongoing harvest a c t i v i t i e s , through the decomposition system. Much of the timber on a given management u n i t can be adeguately covered by the a l t e r n a t i v e management sequences provided by Timber RAM, Only a p o r t i o n of the u n i t would p r o f i t from the more complete s e t s of management seguences a v a i l a b l e through the decomposition scheme. A p r a c t i c a l s t r a t e g y would be to generate a Timber RAM model f o r the whole u n i t , i n c l u d i n g the treatment u n i t s to be i n t e n s i v e l y managed, and opt i m i z e normally. The problem would then be r e s t a r t e d i n decomposition mode, from i t s c u r r e n t optimum, augmented by the network subproblems f o r each of the i n t e n s i v e l y managed treatment u n i t s . S t r a i g h t f o r w a r d and inexpen s i v e LP f i n d s a p o l i c y i n the f e a s i b l e neighborhood of the optimum, before s w i t c h i n g to the more expensive, but more complete, decomposition procedure. A f i n a l a p p l i c a t i o n of the d e c i s i o n a n a l y s i s techniques d e s c r i b e d i n t h i s t h e s i s p e r t a i n s to the problem of i n t e g r a t i n g the management of the timber and non-timber r e s c u r e s of the f o r e s t . The non-timber resources are t y p i c a l l y t r e a t e d as commodities and brought i n t o the planning problem as 175 c o n s t r a i n t s . However, unl e s s a concensus can be formed on d o l l a r value of the resource, i t cannot compete f o r a l l o c a t i o n with the timber h a r v e s t i n g a c t i v i t i e s i n the o b j e c t i v e f u n c t i o n . The decomposition system could be u t i l i z e d t o e v a l u a t e these t r a d e - o f f s . The non-timber resource commodities would be r e p r e s e n t e d i n the LP master problem as c o n s t r a i n t s , and the submodels would simulate the impact of a timber management a c t i o n on the non-timber resource, (Conversely, a management a c t i o n on a non-timber resource might have an impact on the decadal vclume harvested.) The non-timber resource could be represented i n the o b j e c t i v e f u n c t i o n as a weighted g o a l . 176 BIBLIOGRAPHY Adams, D.M. and A, R. Ek. 1576, D e r i v a t i o n s of o p t i m a l management guides f o r i n d i v i d u a l stands. Proc, Soc. Amer, For. Systems A n a l y s i s Working Group. 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Accepted f o r p u b l i c a t i o n . 185 A P P E N D I X I A S u m m a r y o f S y m b o l s a n d N o t a t i o n 1 8 4 L a n d U n i t s t h e m a n a g e m e n t u n i t ( e . g . a P S Y U ) . a t y p e i s l a n d , i E E . a t r e a t m e n t u n i t , u <— E . A t r e a t m e n t u n i t i s a n a g g r e g a t i o n o f t y p e i s l a n d s . T h e r e a r e U t r e a t m e n t u n i t s i n E . T i m e U n i t s t t h e t i m e i n t e r v a l . T h e p l a n n i n g h o r i z o n i s d i v i d e d i n t o T d i s c r e t e t i m e u n i t s ( t = 1 , 2 , . . . , T ) , n o t n e c e s s a r i l y o f e q u a l l e n g t h . M a n a g e m e n t A c t i v i t y a m a n a g e m e n t a c t i o n i n t i m e i n t e r v a l t o n t r e a t m e n t u n i t u . t h e s e t o f a l l f e a s i b l e m a n a g e m e n t a c t i o n s o n u d u r i n g t . u t u t S t a t e s u t ' j u t j u a m a n a g e m e n t s e q u e n c e o r s e t o f m a n a g e m e n t a c t i o n s f o r t r e a t m e n t u n i t u , s c h e d u l e d o v e r t h e t i m e f r o m T . t h e s e t o f a l l m a n a g e m e n t s e q u e n c e s f e a s i b l e o n u . t h e s e t o f a l l m a n a g e m e n t s e q u e n c e s f e a s i b l e o n E . a c a n d i d a t e s e t o f m a n a g e m e n t s e q u e n c e s t o b e a p p l i e d t o t h e t r e a t m e n t u n i t s u C E . t h e s i m u l a t e d m a n a g e m e n t s t a t e o f t r e a t m e n t u n i t u a t t h e b e g i n n i n g o f i n t e r v a l t . t h e a m o u n t o f c o m m o d i t y j p r o d u c e d o r c o n s u m e d o n u d u r i n g t . A c o m m o d i t y m a y b e t i m e s p e c i f i c ( e . g . t h e v o l u m e o f t i m b e r c u t i n t h e s e c o n d d e c a d e ) a n d i n d e p e n d e n t o f t h e a c t i o n t i m e t , t h e t o t a l c o m m o d i t y i n p u t s t a t e , d e f i n e d a s t h e t o t a l c o m m o d i t y s t a t e j u s t p r i o r t o u b e i n g s c h e d u l e d . u - l T C . = Z £ c j u i i t J 1 = 1 t = l 3 1 185 C. the t o t a l commodxty input state when u has been scheduled iu t . . , up to time i n t e r v a l t. u-1 t C. = E C.. + I c. . -jut . . j i 1 juk J i = l k=l M the simulation model applicable to treatment unit u. u { v u ( t + l ) ' C u ( t + l ) } = M u ( { V u t ' C u t } ' a u t ) Commodity Constraints a^ a f e a s i b l e management plan for E. g a constraint on commodity j . Wt> "° Objectives and Returns R return on applying a management sequence to a treatment u t u n i t at time t . R = R (v ,c ,a ) ut u t v ut' ut' ut R management sequence return. T R = R ( v , c , a ) = E R i . u u u u u ^ ut R management unit return function. U R = R(c, a) = E R U=l u 186 APPENDIX II I t e r a t i v e Approximation of F i n a l Conditions Optimization of the multistage model involves the following problem: At stage U of MP1, for each input commodity state C^, f i n d an optimal management sequence a^, such that the output commodity state C w i l l be i n the f e a s i b l e i n t e r v a l {LB.,UB.}. 3 3 i.e. find a y ^ R^Cv^c^jay ) is maximized, and T C.TT + ^ c . _ = C.TT , LB.< C.TT < UB. A2-1 jU t = 1 jUt JU J - JU - 3 for all j = 1, 2, J constrained commodities. To s i m p l i f y the discussion, we w i l l consider only the lower bound and express A2-1 as the i n e q u a l i t y V V S V +j>t " ™ £ ° A2-2 The corresponding Lagrange problem i s to f i n d a and A ,A , ...,A such that U X /. J L(a*,A) = R u ( v u , c u , a u ) + £ \ . g . ( a ^ A2-3 i s a maxxmum. To motivate the algorithm, consider the hypothetical stationary points computed without reference to constraint j (Eq. A2-2), that are plotted on Figure below. 1 8 7 a PI o,o I .. o -P5 X. J - o P2 P3 F i g u r e 2 9 . S e a r c h i n g f o r c o m p l e m e n t a r y s l a c k n e s s , P I : g. >0, A. = 0 J 3 P I i s a n i n t e r i o r o p t i m u m . T h e K u h n - T u c k e r c o m p l e m e n t a r y s l a c k n e s s c o n d i t i o n i s m e t : g.X. = 0 3 3 P 2 : gj,<0,X_. = 0 T h e s o l u t i o n P 2 i s i n f e a s i b l e w i t h r e s p e c t t o t h e p r i m a l p r o b l e m c o n s t r a i n t j . H o w e v e r , t h e c o m p l e m e n t a r y s l a c k n e s s c o n d i t i o n i s m e t : 1 8 8 P 3 : g . < 0 , A . > 0 3 3 T h e s o l u t i o n P 3 i s s t i l l i n f e a s i b l e a n d c o m p l e m e n t a r y s l a c k n e s s d o e s n o t o c c u r : 8 . X . <0 P 4 : g ^ > 0 , A > 0 T h e s o l u t i o n P 4 i s p r i m a l f e a s i b l e , b u t c o m p l e m e n t a r y s l a c k n e s s d o e s n o t o c c u r : g . X . > 0 P 5 : g j = 0 , A > 0 P 5 i s a b o u n d a r y o p t i m u m . T h e c o m p l e m e n t a r y s l a c k n e s s c o n d i t i o n h o l d s . T h e a l g o r i t h m l i s t e d b e l o w t e s t s f o r a n o p t i m u m , i n t e r i o r w i t h r e s p e c t t o c o n s t r a i n t j . I f a n i n t e r i o r o p t i m u m d o e s n o t e x i s t , ^ j > u a r e s e a r c h e d u n t i l t h e c o m p l e m e n t a r y s l a c k n e s s c o n d i t i o n o c c u r s , g . A . = 0 . 3 3 * E v e r e t t ( 1 9 6 3 ) h a s s h o w n t h a t f o r A . f i x e d , i = 1 , . . . . J , i # i , g . ( a T T ) x 3 U i s a m o n o t o n i c a l l y n o n - d e c r e a s i n g f u n c t i o n o f A . C o n s e q u e n t l y , s e a r c h t e c h n i q u e s s u c h a s t h e s e q u e n t i a l s i m p l e x d e s c r i b e i n S e c t i o n 4 . 1 , c a n b e u s e d g u i d e t h e s e l e c t i o n o f \ j > u ' 1 8 9 A l g o r i t h m S t e p : 1 . s e t X. = 0 3 f i n d a u 3 L ( a { j > A J i s m a x i m i z e d i f g.. ( a ^ ) _> 0 t h e n g o t o s t e p 4 o t h e r w i s e , c o n t i n u e 2 . S e l e c t a n e w A . > 0 3 3 . C o m p u t e a ^ 3 L ^ a u ' ^ j ^ i s m a x i m : i - z e d i f g . ( a ^ ) < 0 t h e n A_. i s t o o s m a l l , g o t o s t e p 2 o t h e r w i s e , c o n t i n u e i f g_. ( a y ) > 0 t h e n A_ i s t o o l a r g e , g o t o s t e p 2 o t h e r w i s e , c o n t i n u e i f g j ( a ^ ) = 0 t h e n g o t o s t e p 4 . 4 . a ^ i s o p t i m a l w i t h i n c o n s t r a i n t j , 190 APPENDIX III Distinguishing Points on a Stochastic Surface The objective of the algorithm i s to decide whether the return evaluated at the p o l i c y point x-^  i s s i g n i f i c a n t l y greater than the return at x^. The return function R i s stochastic, and i s computed with a random number e~N(0,d). R = R(x , e.) l j 1 J The algorithm repeats the valuation of the return function with d i f f e r e n t random numbers, i n e f f e c t sampling the stochastic function. Sampling i s repeated u n t i l a s t a t i s t i c a l test of s i g n i f i c a n c e i s passed, or a preset maximum sample s i z e i s reached. On entering the algorithm, the following information must be a v a i l a b l e : the p o l i c y points. the estimated returns for the p o l i c y points. the current sample s i z e that the returns R are based on. the maximum sample s i z e to be allowed f o r either p o l i c y . the standard error of estimate of the return function. the table value of the standard normal d i s t r i b u t i o n above which the one t a i l e d test of the hypothesis HO : R., = R 0 w i l l be rejected. X l ' X2 V R2 " n -max o z 191 Steps 1) set TEST = .FALSE. (R-L " R 2 ) 2) compute z = a + _ L n l n2 3) test of z > Z yes: set TEST = .TRUE. go to step 9 no: continue 4) f i n d the smallest sample i . e . f i n d k such that n^ = minimum (n^, TI ) 5) te s t i f maximum sample s i z e i s reached i f IL > n K — max yes: go to step 9 no: continue 6) simulate p o l i c y x^ with a new random number e + ^ and compute a new average return \ = V k + R ( V \ + l ) ) / n k + 1 7) add new sample to counter n k = \ + 1 8) go to step 2. f .TRUE. R > R 2 9) i f TEST = S > F A L S E - ^ i s n o t s i g n i f i c a n t l y d i f f e r e n t from R 2 END. 192 APPENDIX IV L i s t i n g Of SIMOPT Scanner LISTING OF FILE SIMOPT.SCAN 09:5 i P.M. OCT. 05, 1976 ID=PSYU 1 CONVERSATIONAL SUPERVISOR FOR OPTIMIZATION OF SINGLE STAND MODELS 2 C 2 C 4 C DOUG WILLIAMS 5 C JULY 8 1974 6 C UBC 7 C 8 C 9 ^CONTINUE WITH IOEOCOM RETURN 10 C 11 C LOCAL STORAGE 12 EXTERNAL GOULDtRMEYER,RKILK 13 REAL P(10,50),STEPf10),OS SPAR(5J,1J) .DESV AR(50.10) 14 INTEGER NPR(25)iNVRt25),LISTC( l u0J /4 * l ,7 *2 ,4 *3 ,8 *4 ,3 *5 ,10*6 ,5 *7 ,4 *8 ,3 *9 ,3 *10 ,4 *11 / 15 * ,1 VARPtlOO) .IPARPUOO) 16 LOGICAL TABSW 17 L0GICAL*1 COMLST(IOO),COMAND<10) 18 C 19 C 20 CONSTANTS 21 NWRITE = 0 22 IPLUS = 0 23 LUY = 8 24 LUST = 9 25 C 26 C SET UP CCMMAND LIST 2 7 CALL MQVECl60,'READDISPLA YEDI TSI MUi-AT EOPTIMIZESUBROUTINEWRITESTOPSETFIXFREE' 28 * ,CCMLST) 29 C 30 C READ COMMAND 31 25. CONTINUE 32 CALL FR5ADI'SCARDS*,* STRING:•, CuMA,^Q,10) 33 C 34 C INTERPRET COMMAND 35 CALL FINDC(C0MAND,10,' ',1,1,N3YTE,1OUM,35,45) 36 N8YTE = NBYTE-1 37 35 CONTINUE < 38 CALL F!NDST(C0MLST,60,C0MAND,N3YTE,1.IBYTE,45.45) 39 ICQM = L I STC(I BYTE) 40 GO TO (1000,2000,3000,4000,5000,6000, 7000,8000,9000,10000,11000) ,ICOM 41 45 CONTINUE 42 WRITE(6,200) CCMAND 43 GO TO 25 44 C 45 C 46 C 47 COMMAND : READ 48 1000 CONTINUE 49 C 50 C READ POLICY LABEL AND LOGICAL UNIT 51 CALL F R E A O l , ' S : • * P L A B E L , 4 , ' I i • . L U P Q L l ) 52 C 53 C FIND POLICY LABEL 54 DO 1025 1=1,1C00 55 ILINE = 1*100*1000 56 FINDILUPOLI'ILINS) 57 READ(LUP0L1,100,END=1029) ALABcL 53 IF(ALA8EL.EQ.PLABEL) GO TC 1035 59 1025 CONTINUE 60 1029 CONTINUE 61 WRITS16.210) PLABEL 62 GO TO 25 63 C 64 C READ POLICY BLOCK 65 1035 CONTINUE 66 BACKSPACE LUP0L1 67 READ!LUP0L1,100) ITTLE(J) ,J=l , 10) 68 DO 1045 1=1,1000 69 READ{LUP0L1,110,END=1049) IPAR 70 IPARP(I) = IPAR 71 BACKSPACE LUP0L1 72 READlLUP0L1,1201 PAR I IPAR1,(D5SPARl1PAR,J),J=1,101 73 1045 CONTINUE 74 STOP 75 1049 CONTINUE 76 NP AR = 1-1 77 DO 1055 1=1,1000 78 READ(LUPOL1,110.END=1059) IVAR 79 IVARP(I) = IVAR 80 BACKSPACE LUPCL1 81 READlLUP0L1,1301 VAR (IVAR),1TYPt(IVAR),RNG(IVAR),BL(IVAR1, 82 1 BU( IVAR), (DESVARUVAR, J l , J = l , l u ) 83 1055 CONTINUE 84 STOP 85 1059 CONTINUE 86 NVAR = 1-1 87 WRITE(6,220) PLABEL,NPAR,NVAR 88 GO TO 25 85 C 90 . C 91 C 5 92 COMMAND : DISPLAY 93 2000 CONTINUE 94 WRITS(6,230) (T T L E ( J 1 , J = 1 , 1 0 ) 95 DO 2025 I=1,NPAR 96 IPAR = IPARP(I) 97 WRITE16.240) I PAR,PAR(I P A R ) , t D 6 S P A R U P A R , J ) , J = l , 1 0 ) 98 2025 CONTINUE 99 WRIT6ie,250) 100 DO 2035 1=1,NVAR 101 IVAR = IVARP(I) 10 2 WRITE(6,2603 IVAR,VAR11VAR),I TYPEiIV A R ),RNGlIVAR),BL(IVAR) , 103 1 BU(IVAR),<DESVAR(IVAR,J) , J = l , 6 i 104 2035 CONTINUE 105 GO TO 25 106 C 107 C 108 C 109 COMMAND : EDIT 110 3000 CONTINUE 111 CALL MTSCMDt'$EDIT POLICY",123 112 GO TO 25 113 C 114 C 115 C .116 COMMAND : SIMULATE 117 4000 CONTINUE 118 NP = 0 119 TABSW = .FALSE. 120 CALL FREADI-2, ,6NDLINE',0) 121 CALL F R E A D ( ' * ' , « I : ' , N R E P ) 122 IFINREP -EQ. 0) TABSW = .TRUE. 123 IF(TABSW) NREP = 1 124 IMOD = PAR(6) 125 GO TO 14025,4035,4045,4055,4065) ,IMQJ 126 4025 CONTINUE 127 ITER = 1 128 X = G0UL0(P,10,1) 129 GO TO 4075 130 4035 CONTINUE 131 X = RMEYER(P,10 »13 132 GO TO 4075 133 4045 CONTINUE 134 X = RKILK(P,10,1) 135 GO TO 4075 136 4055 CONTINUE 137 CALL INITL 138 CALL EVCIN 139 REWIND 2 140 X = VCMOLKPf 10, 1) 141 GO TO 4075 142 4065 CONTINUE 143 CALL INITL 144 CALL INITC 145 CALL EVCIN ,_, 146 REWIND 2 ifl 147 REWIND 4 0 1 148 X = VCMDL21P,10,1) 149 4075 CONTINUE 150 X = -X 151 WRITE16.400) X 152 GO TO 25 1 5 3 C 154 C 155 C 156 . COMMAND : OPTIMIZE 157 5000 CONTINUE 158 TABS k = .FALSE. 159 NP = 0 160 C 161 C READ IN NUMBER OF ITERATIONS AND RtPETITIONS 162 CALL FREADI-2,* ENDLINE',0) 163 CALL FREADf•*<, *2 I :'.NITER,NREPJ 164 CALL FREADC - Z t'SNQLINE',* STREAM' ) 165 IF1NRSP .EQ. 0) NREP = 1 166 C 167 C LOAD SIMPLX BUFFERS WITH FREE VARIABLES 168 DO 5 0 2 5 I=1,NVAR 169 IVAR = IVARP(I) 170 IF I I T Y P £ ( I V A R ) . N c . 1 ) GO TO 5025 171 NP = NP+1 172 P(1,NP) = VAR(IVAR) 173 STEP1NP) = RNG<IVARJ 174 IP(NP) = IVAR 175 5025 CONTINUE 176 WRITE(6,410) ( I P f J ) , J = 1,NP) 177 C 173 C HAND APPROPRIATE MODEL TO SIMPLX 179 NN = NP+1 180 IMOD = PAR(6) 181 GO TO ( 5 0 3 5 , 5 0 4 5 t 5 0 5 5 , 5 0 6 5 , 5 0 7 5 ) , I M Q O 182 5035 CONTINUE 133 CALL SIMPLXCF,P,10,NN,STEP,0,NITER,i...0 0 1 ,GOULD,£5035) 184 GO TO 5 0 S 5 185 5045 CONTINUE 186 CALL SIMPLXIF,P,10,NN,STEP,0,NITER,1 . . 0 0 1 ,RMEYER,£5085) 187 GO TO 5085 1 8 8 5 0 5 5 CONTINUE 189 CALL SIMPLX{F,P,1 0 ,NN,STEP,0,NITER,!. . 0 0 1 ,RKILK,£5085) 1 9 0 GO TO 5 0 8 5 1 9 1 5065 CONTINUE 192 CALL S I M P L X C F . P . 1 0 , N N , S T E P , 0 , N I T E R , I , . 0 0 1 , V C M D L 1 , £ 5 0 8 5 ) 1 9 3 GO TO 5 0 8 5 194 5075 CONTINUE 195 CALL S I M P L X ( F , P , 1 0 , N N , S T E P , 0 , N I T E R , 1 . . 0 0 1 , V C M D L 2 , £ 5 0 8 5 ) 196 C 197 COPY NEW VARIABLES BACK TO POLICY 198 5085 CONTINUE 199 00 5055 1=1,NP 200 J = I P ( I ) 201 VAR(J) = P ( l , l ) 202 5095 CONTINUE 203 PROFIT = -F 204 WRITE(6,270) NITER,PROF IT,IP11,1),1 = 1,NP) 205 GO TO 25 206 C 207 C 208 C 209 COMMAND : SUBROUTINE 210 6000 CONTINUE 211 CALL SUBLNK 212 GO TO 25 213 C 214 C 215 C 216 COMMAND : WRITE 217 7000 CONTINUE 213 C 219 C READ LOGICAL UNIT OF OUTPUT POL ICY 220 CALL FREADI'*•,•I:•,LUP0L2) 221 C 222 C WRITE POLICY TO LUPQL2 223 NWRITE = NWRITE + 1 224 NLINE = NWRITE*100C00 225 FIND(LUPCL2'NLINE) 226 WRITE(LUP0L2,100) {TTLEC J ) , J = l , i 0) 227 DO 7025 I=1,NPAR 228 IPAR = IPARP(I) 229 WRIT=(LUP0L2,280) I PAR,PAR{I PAR) , ( D i S P A R t I P A R , J ) , J = l,1 0 ) 230 7025 CONTINUE 231 ENDFILE LUP0L2 232 DO 7035 1=1,NVAR 233 IVAR = IVARP(I) 234 WRITE1LUP0L2.290) IVAR,VAR{IVAR) ,1TYPE< I VAR),RNG<IVAR),BL{IVAR), 235 1BU(IVAR),(DESVARCIVAR,J j,J=1,10) 236 7035 CONTINUE 237 ENDFILE LUPOL2 238 GO- TO 25 239 C 240 C 241 C 242 COMMAND : STOP £ 243 8C00 CONTINUE - j 244 STOP ' 245 C 246 COMMAND : SET 247 9000 CONTINUE-248 CALL F R E A D I « S : • , T , 1 , • I , R : * , 1 N D , V A L N E W ) 249 CALL F I N D C l T . l . ' V ' . l t l . I F I N . I F O , £ 9 0 1 5 1 250 VAR(I ND) = VALNEW 251 GO TO 25 252 9015 CONTINUE 253 PAR(IND) = VALNEW 254 GO TO 25 255 C 256 COMMAND : FIX 257 10000 CONTINUE 258 CALL F R S A D I ' f ' . ' I r ' t l N D ) 259 ITYPS I IND1 = 0 260 GO TO 25 261 C 262 COMMAND : FREE 263 11000 CONTINUE 264 CALL FREAD1•*•,•I:•,IND) 265 I TYPE(IND) = 1 266 GO TO 25 267 C 268 C FORMATS 269 100 FORM AT(20A4) 270 110 F0RMATII3) 271 120 F0RMAT(3X,F6.0,T33,10A4) 272 130 F0RMAT(3X,F6.0,I3,3F6.0,T33,10A4) 273 200 FORMAT (/,'***• ,10A1,' CANNOT 3i£ INTERPRETED ***•,/) 274 210 FORMAT!/, '***' ,A4,• POLICY CANNOT oc FOUND ***',/) 275 220 FORMAT (/,'POLICY ',A4,' ,',I3,« PARAMETERS , ' , I 3 , ' VARIABLES ', 276 1 • ARE READ IN.',/) 277 230 F O R M A T ! / / / , 1 0 A 4 , / , 4 0 ! « . ' ) , / / , ' "ARAMETER ' • , 278 l ' O E S C R I P T I O N',/« # VALUE',/) 279 240 FORMAT(I4,F8.2,7X,10A4) 230 250 FORMAT!//,' VARIABLE',T23,'STEP LOWER UPPER',/,' # VALUE 281 1 'STATUS SIZE BOUND BOUND D t S C R I P T I O N ' , / I 232 260 FORMAT!13,F8.2,16,F9.2,2F7.1,4X.6A4) 233 270 FORMAT!/,'AFTER' ,13, '• IT E RAT IONS , THE BEST RETURN IS ' . F9. 3, /, 5X, • FR 284 1EE V4RIABLES : ',8F7-2,/) 285 280 FORMAT (13,F6.0.T33,10A4) 286 290 F0RMAT(I3,F6-0,I3,3F6.0,T33,10A4) 287 400 FORMAT!/,• OBJECTIVE VALUE IS ',F10.2,/) 288 410 FORMAT!/,T17,'V A R I A B L E S',/,' *,« # Z* ',518) 239 C 290 C 291 END oo 199 Simulated Managed Stand Y i e l d Tables - Meyers POLICY #1 YIELDS PER ACRE OF MANAGED, EVEN-AGED STANDS OF PONDEROSA PINE SITE ilMDiX 70, 20-YEAR CUTTING CYCLE THINNING L£VcLS= INITIAL - 80., SUBSEQUENT - 80. ENTIRE STAND BEFORE AND AFT2* THINNING . PERIODIC INTERMEDIATE CUTS STAND AGE (YEARS) TREES NO. BASAL AREA SQ.FT. AVERAGE D.B.H. IN. 'AVERAGE HEIGHT FT. TOTAL VOLUME •CU.FT. MERCHANT-ABLE VOLUME CU.FT. SAWTIM3ER VOLUME BD.FT. TREES NO. BASAL AREA SQ.FT. TOTAL VOLUME CU.FT. MERCHANT-ABLE VOLUME CU.FT. SAWTIMBER VOLUME BD.FT. 30 30 950 283 119 57 4. 3 6.0 25 27 l i * 0 6J0 300 300 0 0 662 62 560 0 0 40 286 83 7.3 36 1230 930 0 50 50 284 172 107 73 8.3 9.1 45 46 1960 14o0 1680 1320 1800 1800 112 29 500 360 0 • 60 171 97 10.2 52 2070 193 0 4800 70 70 171 104 115 80 11.1 1L.9 59 60 2340 20lQ 2660 1910 9200 7500 67 35 810 750 1700 80 104 96 13.0 65 27U0 2550 10500 90 90 104 67 111 80 14.0 14. 8 70 71 34J0 24dO 3230 2360 14400 10900 37 31 920 870 3500 100 67 92 15.9 75 30aO 2910 14400 110 110 67 21 104 40 16.9 18.6 79 80 3630 14i0 3470 1360 13000 7400 46 64 2220 2110 10600 120 21 47 20.2 34 1740 1680 9700 130 21 54 21. 7 86 20a0 2010 TOTAL 12200 YIELDS 7090 6100 28000 MINIMUM CUTS FOR INCLUSION IN TOTAL YIE L D S — 320. CUBIC FEET AND 1500. BOARD FEET MERCH. CU. FT . - TREES 6.0 INCHES D.B.H . AND LARGER TO 4-INCH TOP. 8D. FT. - TRE es IO.O INCHES D. B.H. AND LARG-R TJ 8-INCH TOP. N o o POLICY #2 YIELDS PER ACRE Oh MANAGED, EVEN-AGED STANDS OF PONDEROSA PINE SITE I N D E X 70, 20-YEAR CUTTING CYCLE STAND AGE (YEA° S) TREES NO. T H I N N I N G L E V J ! L S = I N I T I A L E N T I R E S T A N D B E F O R E A M D A F T E R T H I N N I N G 76., SUBSEQUENT - 100. PERIODIC INTERMEDIATE CUTS B A S A L A R E A SQ.FT. AVERAGE D . B . H . IN. AVERAGE TOTAL MERCHANT-HEIGHT VOLJME ABLE VOLUME FT. CU.FT. C U . F T . ' SAWTIMBER VOLUME BD.FT. TREES NO. BASAL AREA SQ.FT. TOTAL VOLUME CU.FT. MERCHANT-ABLE VOLUME CU.FT. SAWTIMBER VOLUME BD.FT. 30 30 40 50 60 950 2 74 272 270 267 119 54 79 104 126 4.8 6.0 7.3 8.4 9.3 25 27 36 45 51 11*0 600 1170 19iO 2640 280 280 880 1650 2410 0 0 0 1500 4500 6 7 6 6 5 5 9 0 70 70 263 154 146 100 10. 1 10.9 53 59 34d0 24u0 3230 2310 8900 7300 1 0 9 4 6 1020 9 2 0 1600 80 90 90 100 110 110 154 154 103 103 103 32 117 133 99 115 130 50 11.3 12.6 13.3 14.3 15.2 17.0 65 69 70 74 78 80 3230 40u0 30^0 3 73 0 445 0 17,30 3030 3780 2860 3540 4230 1700 12400 15300 12100 16100 20400 8700 51 7 1 34 80 9 8 0 2 6 7 0 9 2 0 2 5 3 0 3 2 0 0 1 1 7 0 0 120 130 32 32 60 69 18.5 19.9 83 36 22J0 2 6 T O 2110 2 540 11500 14600 TOTAL YIELDS 7900 6910 3 1 1 0 0 MINIMUM CUTS FOR INCLUSION IN TOTAL Y I E L D S — 320. CUBIC FEET AND 1500. BOARD FEET MtRCH. CU. FT. - TREES 6.0 INCHES D.B.H. ANJ LARGER TO 4-INCH TOP. BD. FT. - TREES 10.0 INCHES D.B.H. AND LARGiR TO 8-INCH TOP. to o POLICY #3 YIELDS PER ACRE OF MAN4 GEO, EVEN-AGED STANDS OF PONDEROSA PINE STAND AGE (YEARS) 30 30 TREES NC. 950 436 SITE INDcX 70, 20-YEAR CUTTING CYCLE THINNING LEV£LS= I N I T I A L - 115., SUBSEQUENT - 108. ENTIRE STAND 3EF0RS AND AFTE* THINNING BASAL AREA SQ.FT. 119 77 AVERAGE AVERAGE TOTAL MERCHANT-D-3.H. HEIGHT VOLUME ABLE VOLUME IN. 4. 3 5.7 FT. 25 26 CU.FT. 1190 3.>0 CU.FT. 320 320 SAWTIMBER VOLUME BD.FT. 0 0 TREES NO. 514 PERIODIC INTERMEDIATE CUTS BASAL AREA SQ.FT. 42 TOTAL MERCHANT-VOLUME ABLE VOLUME CU.FT. 360 CU.FT. SAWTIMBER VOLUME BD.FT. 40 50 60 432 424 412 109 134. 155 6.8 7.6 3.3 35 44 50 1570 2 4 i 0 3190 1060 1900 2730 0 0 3300 70 70 395 209 171 107 8. 9 9.7 57 58 3 990 25o0 3560 2360 6600 5000 186 64 1430 1200 1600 80 90 90 203 208 133 127 147 108 10. 6 11.4 12.2 64 69 69 3420 4330 S2dO 3190 4060 3030 9800 16000 11700 75 39 1110 1030 4300 100 133 124 13.1 73 397 0 3760 15800 110 110 120 130 133 40 40 40 140 54 64 74 13.9 15.7 17.1 13.4 77 79 82 85 4730 13/0 2320 2760 4480 1780 2220 2 670 20100 8600 11500 14600 93 86 2860 2700 11500 TOTAL YIELDS 8540 7600 32000 MINIMUM CUTS FOR INCLUSION IN TOTAL Y I E L D S — 320. CUBIC FEET AND MERCH. CU. FT. - TREES 6.0 INCHES D.B.H. AND LARGER TO 4-INCH TOP. BD. FT. - TREES 10.0 INCHES D.B.H. AND LARGER TO 8-INCH TOP. 1. BOARD FEET to O to POLICY #4 STAND AGE (YEARS) 30 30 40 50 60 70 70 80 90 90 100 110 110 120 130 TREES NO. 950 '421 417 410 399 384 206 205 205 131 131 131 40 40 40 YIELDS PER ACRE OF MANAGED, EVEN-AGED STANDS OF PONDEROSA PINE S I T E iNDEX 70, 20-YEAR CUTTING CYCLE THINNING LEVeLS= INITIAL - 111., SUBSEQUENT - 108. ENTIRE STAND BEFORE AND AFT E* THUNING BASAL AREA SQ.FT. 119 75 105 133 154 170 108 128 148 108 124 140 54 64 74 AVERAGE D.B.H. IN. 4. 8 5.7 6.8 7. 7 3.4 9.0 9.8 10.7 11.5 12.3 13.2 14.0 15.7 17.1 18.4 AVERAGE HE IGHT FT. 25 26 35 44 50 57 58 69 69 73 77 79 82 85 TOTAL MERCHANT-VOLUME ABLE VOLUME CU.FT. 1190 8 i 0 I5d0 2390 31/0 39/0 25dO 34T0 4350 32 J Q 39d0 4 7J0 1370 23^.0 2790 CU.FT. 310 . 310 1030 192 0 2730 3570 2380 3220 4080 3 040 3770 4490 1780 2220 2680 SAWTIMBER VOLUME BD.FT. 0 0 0 0 3500 6900 ' 5200 10200 16500 11900 16000 20200 8600 11500 14600 TREES NO. 529 178 74 91 PERIODIC INTERMEDIATE CUTS BASAL TOTAL MERCHANT-AREA VOLUME ABLE VOLUME SQ.FT. CU.FT. CU.FT. 44 380 0 62 40 86 1390 1120 2860 1190 1040 2710 SAWTIM3ER VOLUME BO.FT. 1700 4600 11600 TOTAL YIELDS 8 540 7620 3250C MINIMUM CUTS FOR INCLUSICN IN TOTAL Y I E L D S — 3 2 J . CUBIC FEET AND MERCH. CU. FT. - TREES 6.0 INCHES D.B.H. ANJ LARGER TO 4-INCH TOP. BD. FT. - TREES 10.0 INCHES D.B.H. AND LARGER Tu 8-INCH TOP. 1. BOARD FEET o 204 APPENDIX VI Simulated Stand Tables - Goulding __L_C.Y___1 \GE TARIF CU VOL IU ** STAND TA3LS : SITE > 7. > 11. 2 3 4 5 6 7 3 9 .0 20 21.7 213 0 0 92 116 96 44 28 20 4 40 33.5 1066 413 0 28 0 8 20 0 i O 40 33. 5 4G93 2400 0 4 24 20 24 40 16 50 37.0 1198 775 8 8 0 8 i l 50 3 7. 0 5181 3818 0 4 20 4 25 60 39.7 1507 1024 0 4 7 0 0 60 39. 7 5546 4356 7 4 4 70 42.1 1555 1334 0 8 0 70 42. 1 5461 5129 3 90 46.1 7858 7580 0 I ** DBH CLASS 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 29 30 31 32 33 24 12 12 12 4 12 8 8 5 8 15 13 12 16 0 11 7 5 0 3 0 0 8 8 0 0 4 0 4 4 0 3 0 4 0 12 4 4. R0L.LC.Y_32 > 7. > 20 21.7 148 38 32.7 1166 38 32.7 3812 48 36.4 1826 48 36.4 4334 53 39.2 1889 58 39.2 4395 68 41.7 5808 IU ** STAND T ABLE : S ITS 150 DBH CLASS 11. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0 0 200 200 112 56 36 16 148 32 24 24 36 8 40 4 20 0 4 2024 0 24 36 32 64 3 -2 4 24 16 16 16 1048 20 8 16 20 0 16 12 0 12 0 4 3273 0 20 4 44 8 0 8 0 8 12 12 12 12 4 1228 0 15 3 0 15 5 0 0 4 4 4 0 8 0 0 1 3637 1 3 27 3 0 0 4 0 0 8 4 8 12 7 4 4936 1 3 0 27 3 0 0 4 0 0 0 8 4 8 8 to o Cn AGE TARIF CU VOL IU ** STAND TABLE : S i r g 150 ** DBH CLASS 20 21.7 > 7. 149 > 11. 0 2 3 4 0 192 200 5 108 6 56 7 36 3 16 9 10 • 11 12 13 14 15 16 17 18 19 20 21 22 23 24 47 47 36.2 36.2 2183 5471 938 3580 4 24 20 0 0 40 44 4 4 52 4 32 32 0 0 24 20 4 0 3 4 12 16 12 12 57 57 39.0 39.0 2490 5172 1668 3841 0 16 8 20 4 0 24 16 0 32 0 0 12 0 8 8 0 0 4 4 8 4 0 12 4 4 12 8 67 41.5 6876 6004 12 8 0 20 27 0 0 4 4 4 0 4 8 8 8 8 4 to o 207 APPENDIX VII A M u l t i p l e D e c i s i o n Thinning Problem As A Network Problem ( K i l k k i ' s Scots Pine Model). The management o u t l i n e input data s e t f o r the f i r s t demonstration problem d e s c r i b e d i n the t e x t i s l i s t e d below: LISTING OF F I L E DATA.GEN13 11:13 A.M. AUG. 04, 1976 ID=PSYU 1 THESIS EXAMPLES 2 1 1 1. 3 2 2 51. 4 3 30 2. 5 SEND 6 SCOTCH PINE , AGE 7 3 8 9 15 10 9 41 . 41. 11 7 53. 71. 12 6 53. 71. 13 5 53. 71. 14 4 61. 86. 15 3 63. 86. 16 2 65. 86. 17 1 67. 86. 18 10 76. 91. 19 10 93. 20 SEND JULY 2 1974 0.0 KILKKI•S AODdL 0 0 9 0 1 7 6 5 10 0 1 4 3 2 1 0 1 4 3 2 1 0 1 3 2 1 10 0 1 2 1 10 0 1 1 10 0 1 10 0 1 10 0 1 0 0 10 10 Line 1 i s an i d e n t i f y i n g t i t l e . L i n e s 2-4 d e f i n e the time frame of the graph. For example, l i n e 4 d e f i n e s stages 3-30 t o be of 2 years d u r a t i o n . The time frame d e f i n i t i o n i s ended with an e n d - o f - f l i e c a r d . Any number o f management o u t l i n e s may then be d e f i n e d , d e l i n e a t e d by e n d - o f - f i l e cards. L i n e s 6, 7 and 8 are used to i d e n t i f y the treatment u n i t and any parameters necessary t o c a l i b r a t e the s i m u l a t i o n model. L i n e s 9 to 19 d e f i n e the a c t i v i t y s t a t e s and t h e i r f e a s i b l e t r a n s i t i o n s . The order of the 208 data e n t r i e s corresponds to the t a b l e e n t r i e s of the management o u t l i n e r e p o r t i n c l u d e d below and d e s c r i b e d i n the t e x t . The runstream f o r the graph generating program i s simple, r e q u i r i n g only two f i l e assignments. The management o u t l i n e f i l e (DATA. GEN 13) i s assigned to be l o g i c a l u n i t 4, and the generated graphs are w r i t t e n to the s e g u e n t i a l f i l e GFILE13 cn l o g i c a l u n i t 3. # $RUN GENERATE 3=GFILE13 4=DAT A.GEN 13 0 # EXECUTION TERMINATED A f t e r generating the graphs on GFILE 13, r e p o r t s can be produced with a second program. Again, GFILE13 i s assigned to l o g i c a l u n i t 3. A s i n g l e data card i d e n t i f i e s the treatment u n i t and c o n t r o l s the type of r e p o r t . Report o p t i o n 3 was used to c r e a t e the management o u t l i n e r e p o r t and arc l i s t i n c l u d e d below. # $RUN REFORT 3=GFILE13 3 S E N D # EXECUTION TERMINATED 209 TREAT KENT UNIT # 1 : SCOTS PINE, AGE 50 E N T R Y STATE FIRST LAST DURATION COSTS VOLUME SIM MIN MAX I AREA VOL I CUT J I OUTPUT STATES 15 J 1 1 I 1 I j F | 9 9 I 11 41 | | I | rr | 7 6 5 10 7 I 53 71 j | I I I I 4 3 2 1 6 l 53 71 I | I j T | 4 3 2 1 5 J 53 71 | \ I | T ! 3 2 1 10 4 I 61 86 | | | l T | 2 1 10 3 ! 63 86 | J I I I I 1 10 2 ! 65 86 | j | I T I 10 1 I 67 86 | | I | T I 10 10 | 76 91 | f l | T j 10 THE CORRESPONDING NETWORK HAS 29886 FEASIBLE MANAGEMENT SEQUENCES AND 1972 ARCS ARC LIST F R O M STAGE STATE A T STAGE STATE 1 15 | 2 9 T 2 9 f 3 5 T 2 9 } 3 6 1 T 2 9 I 3 7 T 2 9 I 4 5 T 2 9 1 4 6 T 2 9 1 4 7 T 2 9 ! 5 5 T 2 9 1 5 6 T 2 9 1 5 7 T 2 9 1 6 5 T 2 9 ! 6 6 T 2 9 1 6 7 T 3 6 1 7 4 T 4 6 1 7 4 T 5 6 1 7 4 T 6 6 | 7 4 T 3 7 1 7 4 T 4 7 1 7 4 T 5 7 1 7 4 ! T 6 7 | 7 4 T 2 9 1 7 5 ! T 2 9 I 7 6 I T 2 9 1 7 7 1 T 3 C 1 8 3 1 T SIMU-LATE? C O S T S VOLUME 210 A d d i t i o n a l treatment u n i t r e p o r t s can be generated or e x e c u t i o n terminated with an end f i l e command. A runstream f o r the network s o l u t i o n program (described i n the t e x t . S e c t i o n 4.2.1) i s l i s t e d below. The graphs are read o f f GFILE13. l o g i c a l u n i t s 3 and 9 are assigned to f i l e s used i n the LP decomposition model d e s c r i b e d i n S e c t i o n 5. At the end of the s o l u t i o n process, f i l e -POL c o n t a i n s a summary of the opt i m a l management seguence f o r each treatment u n i t , at f i l e -LP c o n t a i n s the same i n f o r m a t i o n expressed as Timber EAK LP v e c t o r s . The parameter »SOLO» s i g n i f i e s t h a t the s o l u t i o n program i s to be run without l i n k i n g to the LP program. # $BUN SOLVE 3=GFILE13 8=-P0L 9=-LP PAE=S0L0 1 1 2 0.02 SEND # EXECUTION TERMINATED A s i n g l e data card c o n t r o l s the s o l u t i o n o f each treatment u n i t graph. The f i r s t f i e l d d e f i n e s the treatment u n i t i n GFILE to be loaded and s o l v e d . The second f i e l d d e f i n e s the o b j e c t i v e f u n c t i o n (maximize MAI, ENw, e t c . ) . The t h i r d f i e l d c o n t r o l s the type of r e p o r t s generated. The f o u r t h f i e l d i s the d i s c o u n t r a t e to be used i n the present net worth c a l c u l a t i o n . For the demonstration problem, the graph loaded from GFILE was the f i r s t (and only) one, The o b j e c t i v e was to maximize the present net worth at a 2% i n t e r e s t r a t e . A stage r e t u r n r e p o r t and a d e c i s i o n s matrix were generated and are i n c l u d e d below. A d d i t i o n a l graphs may be loaded and s o l v e d , or execution terminated with an e n d - f i l e command. The stage r e t u r n r e p o r t l i s t s v a r i o u s values of i n t e r e s t 211 f o r each s t a t e and stage combination. The optimal s t a t e at each stage i s a l s o g i v e n . Five values are recorded a t each f e a s i b l e s t a t e - s t a g e combination. the maximum stage r e t u r n c a l c u l a t e d with the LP commodity values ( c o n s t r a i n e d d e c i s i o n d e r i v a t i v e ) , the o b j e c t i v e stage r e t u r n on e n t e r i n g the s t a t e , the volume produced on e n t e r i n g the s t a t e , the c a p i t a l produced (or consumed) on e n t e r i n g the s t a t e , the maximum stage r e t u r n (unconstrained d e c i s i o n d e r i v a t i v e ) . The d e c i s i o n s matrix records f o r each s t a g e - s t a t e combination, the optimal s t a t e and stage that the system should have entered from. The d e c i s i o n s matrix can be used to t r a c e back the op t i m a l management seguence from any ( f e a s i b l e ) stage and s t a t e . 1 OPTIMAL 15 S T A G E . R E T U R N S . 4 5. a 2 -lfl. .11. . 1 2 . . 1 3 . . 1 4 - .15.. 0.?. 7 0.27 1.25 0.78 0.2 7 1 .03 0. 0.83 0.52 27 27 1.29 0.81 0.27 1.05 0.6t> 0.85 0.54 0.27 1.08 0.69 0. 87 0.55 0.27 0.27 1.11 0 . 7 i 0. 89 0.57 0.81 0.54 0.27 0.27 1.12 0.72 0.89 0.57 0.27 0.74 0.52 0.27 0.83 0.56 0.27 0.27 1.13 0.75 0.90 0.59 to to 1 0.27 1 1.44 0-80 0. 86 1-14 0.91 1 1 .04 0.57 0.61 0.77 0.62 1 0.55 0.30 0.27 0.27 11 1 5 0.55 0.30 0.27 0.27 | 0.28 | 1.49 0. 82 0.87 1.13 0.90 j 1.09 0.60 0.62 0.77 0.62 1 0.55 0 . 3 0 0.27 0.27 12 1 5 0.55 0.30 0. 27 0.27 | 0 . 23 | 1.53 0-8 3 0.88 1.13 0.90 | 1. 14 0.61 0-64 0.79 0-63 1 0.55 0.30 0.27 0.27 1 3 1 1 0.55 0.30 0.27 0.27 | 0. 28 | 1.53 0.35 0.83 j 1.19 0.64 0.65 1 0.5 5 0.30 0.27 0.27 1.4 1 1 0.55 0.30 0.27 0.27 0.82 0.27 | 0.36 0.90 1.45 | 0.65 0.67 1.18 I 0.55 0.30 0.27 0.27. 0. 82 15 1 1 0.55 0.48 0.27 0.27 0.83 | 0.20 0.28 j 1.21 0.87 0-89 1.56 | 0.93 0.67 0.66 1.29 1 0 . 55 0.48 0.27 0 . 2 7 0.83 1 6 1 1 0.55 0.48 0.27 0.27 0.84 | 0.29 | 0.87 0 . 9 0 - . 1 . 6 6 j 0.67 0 . 6 3 1.39 1 0.55 0 . 4 8 0.27 • 0 . 2 7 0.84 17 I 1 0.55 0.48 0.27 0.27 0.85 | ' 0 . 3 0 0.38 0 . 8 9 1.77 | 0.69 0.68 1.50 1 0.55 0 . 4 8 0.27 0 . 2 7 0.85 1 8 1 1 0.55 0.48 0.2 7 0.27 0.66 0-31 | 0.39 0 . 8 9 1.88 | 0.70 0 . 6 8 1-61 0.55 0 . 4 3 0.27 0 . 2 7 0.86 2 1 4 ( ^ O CO ro —i CO CO o> ro o <!• o co rg o CO O O N H O r-r j rvj CO -1-0 ^ ^ in o co ro ro o co o o r g N O CO <7> CO ro ___1_I__S.__AT_I_ FROM { STATE, STAGE ) _IA._E.___1 7 _, 1 2 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 2 4 0 0 0 0 0 0 0 0 9 3 5 0 0 0 0 0 0 0 0 5 4 6 0 0 0 0 0 0 0 0 5 5 7 0 0 0 0 0 0 7 6 5 6 8 0 0 0 0 5 7 7 7 • 5 7 9 0 0 7 3 5 8 7 3 5 8 10 5 9 2 9 5 9 7 9 5 9 11 5 10 2 10 5 10 7 10 5 10 12 5 11 2 11 5 11 7 11 5 11 13 5 12 2 12 5 12 . 7 12 5 12 14 1 13 2 13 5 13 7 13 5 13 15 1 14 5 14 5 14 7 14 5 14 16 1 15 2 15 5 15 7 15 5 15 17 1 16 2 16 5 16 7 16 5 16 18 1 17 ? 17 5 17 7 17 5 17 19 . 1 18 2 18 5 18 7 18 5 18 20 1 19 2 19 3 19 4 19 5 19 21 1 20 2 20 3 20 4 20 5 20 22 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 S T A T E S. 0 0 0 o • 0 0 15 1 0 0 9 2 9 2 0 0 9 2 0 0 9 3 9 3 0 0 9 3 0 0 9 4 9 4 0 0 9 4 0 0 9 5 9 5 0 0 9 5 0 0 9 6 9 6 0 0 9 6 0 0 9 7 9 7 0 0 9 7 0 0 9 0 9 0 0 9 8 0 0 9 9 9 9 0 0 9 9 0 0 9 10 9 10 0 0 9 10 0 0 9 11 9 i l 0 0 9 11 0 0 6 12 7 12 0 0 9 12 0 0 6 13 7 13 0 0 9 13 1 13 6 14 7 14 0 0 9 14 1 14 6 15 7 15 0 0 9 15 1 15 6 16 7 16 0 0 9 16 1 16 6 17 7 17 0 0 9 17 1 17 6 18 7 18 0 0 9 13 1 18 6 19 7 19 0 0 9 19 1 19 6 20 7 20 0 0 9 20 1 20 0 0 0 0 0 0 0 0 1 21 0 0 0 0 0 0 0 0 , 10 22 .11 12 1_ 14 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 ' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Ln 216 The second m u l t i p l e t h i n n i n g problem d i s c u s s e d i n s e c t i o n 4.2.3 allows up t o t h r e e t h i n n i n g s . Two t h i n n i n g l e v e l s i n c l u d e d i n the pre v i o u s problem were not co n s i d e r e d (DL = 1.5 corresponding to s t a t e 4, and DL =3.0 corresponding to s t a t e 7 ) , but the entry times f o r the second t h i n n i n g s t a t e s were g e n e r a l l y reduced by 10 years.. Consequently, the r e s u l t i n g graphs are about the same s i z e f o r the f i r s t and second problem, 29886 a r c s and 29729 a r c s , r e s p e c t i v e l y . The s t a t e s are d e f i n e d as before (see F i g u r e 11). The precedence graph (Figure 3 0 ) and the management o u t l i n e r e p o r t are i n c l u d e d below. F i g u r e 3 0 . P r e c e d e n c e g r a p h f o r K i l k k i ' s d e m o n s t r a t i o n p r o b l e m . 217 # E EEPOET 3=GFILE12 # EXECUTION BEGINS 3 TREATMENT UNIT J 3 : SCCIS PINE E N T E V DURATION COSTS VOLUME SIM STATE FIRST LAST| MIN MAX |ARJ! VOL | CUT | | OUTPUT STATES 1 5 I j j I | F ! 9 9 I 4 1 4 1 | I I I T I 6 5 1 0 6 1 5 1 7 1 | I 1 1 T 3 2 1 1 0 5 1 5 1 7 1 | | 1 | T 3 2 1 1 0 3 1 5 3 8 6 J | 1 1 T 1 1 1 0 2 1 5 5 8 6 \ | 1 1 T 1 0 1 | 5 7 8 6 | | I 1 T 1 0 1 0 | 6 1 1 0 1 | | 1 1 T THE CORRESPONDING NETwOEK HAS 29729 FEASIBLE MANAGEMENT SEQUENCES END OF FILE # EXECUTION TEBMINATED The runstream to s o l v e the network problem f o r 7 d i s c o u n t r a t e s i s l i s t e d below. The present net worth was c a l c u l a t e d with a 5 year r e g e n e r a t i o n l a g between r o t a t i o n s . $E SOLVE 3=GFILE12 8=-LP S=-POL PAB=S0L0 3 1 1 0. 02 5. 3 1 1 0. 03 5. 3 1 1 0.G4 5. 3 1 1 0. 05 5. 3 1 1 0. 06 5. 3 1 1 0. 07 5. 3 1 1 0. 08 5. $END JB *LIST DUGG:DATA.GEN 12 -POL -LP END 218 APPENDIX VI I I A M u l t i p l e D e c i s i o n T h i n n i n g Problem As A Network Problem (Goulding's Douglas F i r Model). A s e t of a l t e r n a t i v e management seguences f o r s i t e 170 Douglas f i r was generated as a d i r e c t e d graph, f o l l o w i n g the management o u t l i n e i n c l u d e d below: ISTING OF F I L E DATA. GEN 14 1 THESIS EXAMPLE : THREE ST0CKI 2 1 2 1. 3 3 3 28. 4 4 30 2. 6 SEND 7 DOUGLAS FIR. SITE 170 8 1 170 200 1.0 F 9 0. 10 15 1. 1. -11 1 2. 2. -12 2 2. 2. 13 3 2. 2. * 14 4 32. 42. • 15 5 42. 52. • 16 6 32. 42. • 17 7 42. 52. -18 8 32. 42. 19 9 42. 52- • 20 10 52. 75. • 22 SEND 11:26 P.M. OCT. 05, 1976 LEVELS - TwO THINS ID=PSYU m 0 0 1 2 3 50. m 0 0 4 5 10 60. . 0 0 6 7 10 70. • m . 0 0 8 9 10 61. 0 1 5 10 61- - 0 1 10 61. . 0 1 7 10 61. . 0 1 10 61. . 0 1 9 10 61. . 0 1 10 • 27- . 0 1 S t a t e s 1 - 3 correspond t o i n i t i a l stand d e n s i t y l e v e l s at age 20, of 600, 900 and 1200 t r e e s per acre, r e s p e c t i v e l y . S t a t a s 4 and 5 repr e s e n t t h i n n i n g s of 30% and 25% of the stand basal area, and can only be entered a f t e r s t a t e 1 (1200 t r e e s per acre a t age 20). S i m i l a r l y , s t a t e s 6 and 7 are t h i n n i n g s of 25% stand b a s a l area a f t e r s t a t e 2, and s t a t e s 8 and 9 are t h i n n i n g s o f 25% and 20%, a f t e r the stand has been i n s t a t e 1. In every case. 219 the cut i s d i s t r i b u t e d across the stand i n the same manner. In terms of diameter c l a s s e s d e f i n e d about the stand average diameter at breast height (DBH) , 50% of the b a s a l area removed i s taken i n t r e e s of dbh fee 1.50 DBH, 30% from |2d1.25, 1.5), and 20% from |2d1.0, 1.25). State 10 i s the c l e a r c u t . Costs of $50, $60, and $70 per acre were i n c u r r e d on e n t e r i n g the s t a t e s of 600, 900 and 1200 t r e e s per ac r e , r e s p e c t i v e l y . Thinning c o s t s of $61 per c u n i t and c l e a r c u t c o s t s of $27 per c u n i t were a l s o charged. A runstream i s l i s t e d below t h a t f i n d s the management seguence with the op t i m a l present net worth, f o r d i s c o u n t r a t e s of .02, .01, .06 and .08. The y i e l d t a b l e s and stand t a b l e s f o r the optimum management seguences f o l l o w : SHUN SOLVE 3=GFILE14 8=LP 9=FGLICY PAB=SOLO 1 1 1 0.02 1 1 1 0.01 1 1 1 0.06 1 1 1 0.08 $$END MAIN 1 C F C P S I 170 A F T F P T H I N N I N G * • G R O S S * * P R O D U C T ION * T H I N N ! N G S " AV T O T A L * T O T A L * ~ ' " FIV. T 0 T A L -A G F S T E M S D * H " PA V O L * V O L MAI * S T E M S CfiH BA VOL D I S C O U N T P A T E : . 0 2 2 0 1 2 0 0 3 . ? 1 0 1 . 1 6 0 1 • * 1 6 0 1 . 80 * 3 9 6 7 3 9 2 8.A 1 6 1 . 5 6 4 3 . 2 2 4 1 3 . 3 2 4 1 . 1 1 5 7 4 . * 8 4 5 4 . 2 1 4 * 1 4 3 8 5 . 2 1 3 0 1 S C O U N T F A T E : . 0 4 2 0 12<"10 3 . 8 1 0 1 . 1 6 C 1 . * 1 6 C 1 . 8 0 * 3 1 5 7 2 6 . 5 1 4 5 . 4 1 6 2 ^ * 6 1 5 6 . 1 9 5 * 6 5 2 4 4 1 3 . 5 2 7 C . 1 2 8 5 3 . * 1 4 8 4 7 . 2 2 7 * D I S C C U N T P AT F : . 0 6 2 0 1 2 0 0 3 . 8 1 0 1 . 1 6 0 1 . * 1 6 0 1 . 3 1 5 8 8 6.5 1 4 7 . 4 2 2 2 . * 6 2 3 5 . 6 7 2 2 3 1 4 . 1 2 7 4 . 1 3 2 6 2 . * 1 5 2 6 5 . 8 0 * 1 9 8 * 2 2 6 * D I S C C U N T P A T E : .C8 2 0 1 2 0 0 3 . 6 1 0 1 . 1 6 C 1 . * 1 6 0 1 . 3 1 5 4 8 "'"6.6 1 4 0 . 4 0 5 1 . * 5 9 5 2 . 3 9 4C8 7 . 6 1 4 7 . 5 0 6 9 . * 8 7 8 3 . 51 3 3 ? 1 0 . 1 2 1 5 . 8 9 8 1 . * 1 2 6 9 6 . 80' * 1 8 9 *" 2 2 2 * 2 4 7 * 9 6 1 1 . 8 7 5 . 2 8 1 1 . 1 3 2 9.3 6 4 . 1 9 9 4 . 1 3 6 9.2 6 4 . 2 0 C 3 , 1 2 8 S.2 8 0 1 C . 5 6 1 . 1 9 C 1 . 5 0 . 1 8 1 3 . AGS' T A F I F CU VCL II) ** STAND TABLE :SITE 170 ** ^ DRH CLASS 20 22.9 C C 152 432 348 144 92 .32 40 40 36. 1 36. 1 265? 4413 1805 1469 8 24 56 38 0 64 24 28 3 44 8 28 20 20 C 26 28 4 8 63 45.4 1C928 3926 12 3 e 24 24 16 24 4 16 16 20 8 0 12 16 16 AG? TARIF CU VCL > 7. IU > 11. 2 3 4 ** 5 STAND 6 TA 7 SLE 8 .SITE 170 9 10.11 12 CBH 13 14 CLASS 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3C 31 32 33 20 22.9 0 0 152 432 348 144 92 32 3 2 31.8 32 31.8 182 1 2C76 415 373 0 4 43 136 0 172 40 60 4 72 36 4 40 16 44 C 8 20 66 44.8 12179 9507 0 8 8 40 28 32 0 20 24 24 8 0 4 16 12 0 ~ 8 12 A.-C T . P I I : r i l U D L IU • ** STAND T A 8L E :SITE 1 7 0 DBH CLASS ... •_. T _ P T . C U V O l ^ I L ^ ...... „ _ 5 ^ . „ S G I Q 1 3 ^ 1 5 1 6 ; 7 1 8 1 9 2 0 21 2 2 2 3 2 4 2 5 2 6 2 7 2 3 2 9 3 0 3 1 32 3 3 2 0 2 2 . 9 C 0 1 5 2 4 3 2 3 4 8 1 4 4 9 2 3 2 22 31.8 1S24 412 0 48 C 40 0 40 8 32 21.3 2163 271 0 8 64 136 1 56 60 8C 16 48 0 20 68 45.4"12536 10462 1 2 C 2 0 24 28 0 20 28 20 8 0 16 20 8 0 0 4 16 AGE TAR IF CU VCL > 7. IU > 11 . 2 3 4 ' 5 STAND TABLE 6 7 8 :S ITS 9 10 170 11 ** 12 D EH 13 14 CLASS 15 16 17 18 19 20 21 22 23 24 25 2o 27 23 29 30 31 32 33 20 22.9 C C 152 432 348 144 92 32 2 2 21.8 1722 346 0 44 0 32 a 40 4 32 31.8 2116 44C 0 16 36 124 163 52 68 2C 4C 0 24 40 36.1 16<?0 714 24 4 16 8 16 8 2C 12 12 12 '.. . . . .. .. 40 36.1 """3 502" 1444 0 12 12 36 """"84 124" "48 12 28 "0 0 52 40.7 7969 5C58 0 4 36 44 68 68 20 0 24 16 16 0 0 0 12 0 0 16 3 to • to 222 a l l the management seguences s e l e c t a d e n s i t y of 1200 stems per a c r e , a l l o w i n g the h e a v i e r t h i n n i n g s (30% of stand b a s a l area followed by 25% f o r a subseguent c u t ) . At the lowest d i s c o u n t r a t e of .02, a s i n g l e t h i n i s scheduled at 39 years. Twenty-six c u n i t s of c l o s e u t i l i z a t i o n volume wood i s removed, with an average dbh of 11.2 inches. At d i s c o u n t r a t e s g r e a t e r than .02, the i n i t i a l t h i n n i n g i s performed a t 31 y e a r s , and the volume and average dbh of the y i e l d d e c l i n e s to 18 c u n i t s and 9.2 i n c h e s , r e s p e c t i v e l y . A second t h i n n i n g i s scheduled at 39 years when the d i s c o u n t r a t e reaches .08. This second t h i n n i n g y i e l d s about 17 c u n i t s of c l o s e u t i l i z a t i o n wood, with an average dbh of 10.5 i n c h e s . The high discount r a t e a l s o f o r c e s down the age when the stand i s c l e a r c u t , from 67 to 51 years. 223 i l P J U J I X IX A Wolfe-Dantzig Decomposition Program In MPSX PROGRAM('ND') * THIS PECGBAM PEEFORMS A PRIMAL OPTIMIZATION WITH * WOLFE-DANTZIG DECOMPOSITION * DOUG WILLIAMS * APRIL 1974 * UBC * * INITIALZ TITLE(•THESIS DEMONSTRATION') * * SUBIN READS IN THE PROBLEM NAME AND OEJECTIVE, AND * SETS UP THE SUBPROBLEM * SUBIN (XPENAME,XCE.3,RATE,ELAG,REPSW,MAXIT,ITER,OBSW ,IOBJ, OPTSW,OBJPSW,RHSPSW,XCHCOL,XCHROW,XPARMAX,XPARDELT) MOVE(XDATA,* TIMERAM') MOVE (XRHS, 'Z') CONVERT PROBLEMS(*PROEFILE•) SETUP ('MAX') MOVE(XDAT2,•TPUNCH') IF (0BSW,0PT1) INSERT OPT1 CRASH PRIMAL NEWZ = XFUNCT XPARAM = 0. MOVE(XOLDNAME,'TIMBRAM2') SAVE (* NAME','SAVEEASF') MOVE(OPZSTART,'CONTINUE*) IF (OBJPSW,PAROEJ) IF(RHSPSW,PARRHS) * * START DECOMPOSITION PHASE * DECOM SELECT ('PI',* VOLU1•,V1#'VOLU2' , V2 , 'VOL U3',V3,•V0LU4*,V4, •VCLU5',V5,»VOLU6',V6,•VOLU7',V7,'V0LU8',V8,* V0LU9',V9, 'VOLUA',VA,«PNR1»,D1 #'PNR2»,D2,'PNR3',D3,•PNR4•,D4,«PNR5•, 224 D5,'PNR6»,D6,•PNR7*,D7,»PNR8' ,D8, 1PNR9',D9,« PNRA«,D A , ' •) FREECORE SOBDC (D1,B2,D3,D4,D5,D6,D7,D8,D9,DA,V1,V2,V3,V4,V5,V6, V7,V8,V9,VA,RATE,RIAG,REPS\<J,OPTFLAG, ITER , XDATA, LEI IE 8 LFI1E9,IOEO,XPARAM) IF (OPTFLAG,OOTP0T) CPTFLAG = .TRUE. REVISE(•FILE»,* REV *,* NOPRINT *) SETUP ('MAX') RESTORE('NAME','SAVEBASE') SELECT(* COL *,'HC1»,H1,»HC2',B2,•HC3',H3,»HC4«,H4, fHC5»,H5, ' HC6•,H6 , 'HC7 1,K7,«HC8«,H8,' HC9 *,H9,'HCA',HA,«HR1',P1, •HR2,,P2,' HR3 *,P3,* HR4 *,P4,'HE5*,P5,* HR6•,P6,• HR7 *,P7, •HR8 ,,P8, ,HR9 ,,P9,*HRA ,,PA,' «) XPARAM = OLDP CONTINUE OPT2 CLDZ = NEWZ OPTIMIZE NESZ = XFUNCT DIFFZ = NEwZ - OLDZ 1F(DIFFZ .IE. .001 ,OUTPUT) SAVEB IF (ITER .EQ. MAXIT ,OUTPUT) OLDP = XPARAM SAVE('NAME*,*SAVEEASE') GOTO (DECCM) * * OUTPUT SECTION * OUTPUT MOVE(XDATA,'TPUNCH') SELECT{«COL»,•BC1 ,,H1,•HC2«,H2,*HC3•,H3,•HC4•,H4,*HC5•,H5, *HC6',H6,»HC7»,B7,»HC8',H8,«HC9',H9,» HCA« ,HA,»HB1 1,P1, •HR2«,P2,«HR3«,F3,«HR4«,P4,•HR5»,P5,•HR6•,P6,»HB7 ' ,P7, VHB8« ,P8, 'HR9» ,P9, »HBA» ,PA, » •) PUNCH('VALUE',«LIST«) SOLUTION IF (OPTSW,STOP) IF (ITER . GE. MAXIT,STOP) CONTINUE * * OBJECTIVE PARAMETRICS FOR GOAL PROGRAMMING WEIGHTS * PAROBJ MVADR(XDGPRINT,SAVEB) MVADB (XDGNMX,OUTPUT) PABAOBO GOTO (STOP) * * RIGHT HAND SIDE PARAMETRICS * PARBHS MVADB (XDGPEINT,STOP) MVADB (XDCPBIM,OPT2) FABARHS GOTO (STOP) * * VOLUME DUAL VARIABLES V1 DC{0.) V2 DC(0.) V3 DC (0.) VU DC (0, ) V5 DC (0.) V6 DC (0.) V7 DC(0.) V8 DC (0. ) V9 DC (0.) VA * DC (0.) * CAPITAL DUAL VSRIA: D1 DC(0.) D2 DC (0.) D3 DC (0.) D4 DC (0.) D5 DC (0.) D6 DC (0.) D7 DC (0.) D8 DC (0.) D9 DC (0,) DA DC (0.) * DECADE HARVEST H1 DC(0.) H2 DC{0.) H3 DC(0.) EH EC (0. ) H5 DC(0.) H6 DC (0.) H7 DC (0.) H8 DC (0.) H9 DC(0.) HA -it DC (0.) * DECADE NET REVENUE P1 DC(0.) P2 DC (0.) P3 DC(0.) P4 EC (0. ) P5 EC (0.) P6 DC (0.) P7 DC (0.) P8 EC (0.) P9 DC (0.) PA # EC (0.) * ICC AI STORAGE LFILE8 DC(1000) LFILE9 DC (1000) OLDZ DC (0. 0) NEHZ DC (0.0) DIFFZ EC (0,0) RATE DC (0.0) R LAG DC (0.0) OL-DP DC (0.0) IOBJ DC (0) ITER DC (0) 226 HAXIT OBSW DC (0) DC (.FALSI.) REPSW DC (.FALSE.) OPTFLAG DC (.FALSE.) OBJPSW DC (.FALSE.) fiHSPSW DC (.FALSE.) OPTSW DC (.FALSE.) * END OF RON * STOP EXIT FEND APPENDIX X Treatment Onit Management O u t l i n e s For Decomposition Demonstration Problem I--_I_E_1____L-__1 : TO! - OVERM4TURE E N T R Y DURATION COSTS VOLUME SIM S-AIEl£IB_I.L_SIl__L____._lAaE___QL_l--,j:i_L_iELAfiL QUteUI <I_T_S I I I I i I 14 I 1 1 1 15 | 2 2 1 | 9 I 3 20 I 1 30.0 8 1 3 20 | 124.0 7 I 3 20 | 1 10.0 6 1 40 100 I | 4 I 50 100 I | 2 1 40 100 I j I 1 60 100 | | 10 | 80 120 I | -5.0 270.0 F 1 15 F 1 9 8 7 F 1 6 4 10 F 1 4 2 10 F 1 1 10 T 1 4 1 10 T 1 1 10 T I 10 T i 10 T 1 THE CORRESPONDING NETWORK HAS 6230 FEASIBLE MANAGEMENT SEQUENCES AND 795 ARCS I _ _ _ I _ E _ I _ _ _ I I _ _ _ _ : T02 - OVERMATURE, DISTANT E N T R Y DURATION CO -E-IEiEI_SI_LASIl__lH__A__l__EA 14 1 1 1 1 1 15 | 2 2 1 | 9 1 3 30 I 135.0 3 I 3 30 I 129.0 7 I 3 30 I 1 10.0 6 1 40 100 I 4 I 50 100 | | 2 1 40 100 I | 1 1 60 100 | 10 I 80 120 I | TS VOLUME SIM _QL_l__£UT_leLAGL_QULP.UI_SI_rPS I I i 1 1 F 1 15 5.01 310.01 F 9 8 7 I 1 F 1 6 4 10 1 1 F | 4 2 10 1 1 F 1 1 10 3.0| | T I 4 1 10 3.01 1 T 1 1 10 3.01 | T i 10 3.01 | T I 10 2.0| | T I THE CORRESPONDING NETWORK HAS 8695 FEASIBLE MANAGEMENT SEQUENCES AND 1040 ARCS I_£_I_E_I_U_II_JK_3 : T03 - OVERMATURE, CURRENTLY INACCESSIBLE E N T R Y DURATION COSTS VOLUME SIM I 14 I 1 1 15 1 10 10 9 1 15 ' 40 8 1 15 40 7 I 15 40 6 1 40 100 4 1 50 100 2 1 40 100 1 1 60 100 10 1 80 120 30.0 24.0 10.0 -5.0 I 270.01 I 15 9 6 4 1 1 10 10 8 4 2 10 1 10 7 10 10 10 THE CORRESPONDING NETWORK HAS 4246 FEASIBLE MANAGEMENT SEQUENCES AND 597 ARCS iaE_I_E_I_lI_T_____ : T04 - AGE 20, HIGH DENSITY E N T R Y DURATION COSTS VOLUME SIM _I_IEI£ia_I_L_ail__I______iAaEA__QL_l__CUI_lELaGl.QULEUI_Sj:_I£S I 15 I 9 I 6 I 4 I 2 I 1 I 10 I 1 2 30 40 50 50 60 I 1 I 2 I 80 \ 80 I 30 | 80 I 110 I 9 6 4 2 10 10 4 1 10 2 10 10 THE CORRESPONDING NETWORK HAS 316 FEASIBLE MANAGEMENT SEQUENCES AND 151 ARCS I_EJI_E_I_ii_II____ : T05 - AGE 35, HIGH DENSITY E N T R Y DURATION COSTS VOLUME SIM SlAIElEI_SI_LASIl__X„__A__l__£A„VQL_l__CUI_iELA_L_QUr.EUI_aiAI£S 15 I 1 I 9 1 2 2 6 1 15 55 4 1 20 60 2 1 30 60 1 1 20 60 10 I 40 100 t 1 F » 1 9 1 F 1 6 4 2 1 T 1 4 1 10 ! T 1 2 10 1 T 1 10 1 T 1 10 1 T 1 10 THE CORRESPONDING NETWORK HAS 1593 FEASIBLE MANAGEMENT SEQUENCES AND 373 ARCS I_E_I_E_I___It____ : T06 - AGF. 50, MEDIUM OEIISITy E M T R Y DURATION COSTS VOLUME SIM SI_IElEIE._I_L4SIl__i______l_aEA_VaL_l__CUI_lEL_GL_Q_CeUL_S.IAIES I I I I I I 15 1 1 1 1 1 1 1 F | 3 8 1 2 2 1 1 1 1 F 1 5 3 10 5 1 3 40 1 I 1 I T I 3 1 10 3 1 4 40 I I 1 I T I 10 1 1 4 40 | | 1 I T I 10 10 1 15 50 I | 1 I T I THF CORRESPONDING NETWORK HAS 2331 FEASIBLE MANAGEMENT SEQUENCES AND 805 ARCS ISEAI_EiJI_!J_II___2 : T07 - AGE 50, MEDIUM DENSITY E N T R Y DURATION COSTS VOLUME SIM SIAT__L£I__T_LASLL__I___A__U^ I I I I I I 15 I 1 1 1 1 1 1 F I 3 8 1 2 2 1 1 1 1 F 1 5 3 10 5 1 3 40 | 1 2.01 1 T 1 3 I 10 3 1 4 40 I 1 2.01 1 T 1 10 1 1 4 40 I 1 2.01 1 T 1 10 10 I 15 50 I 1 2.01 1 T 1 THE CORRESPONDING NETWORK HAS 2331 FEASIBLE MANAGEMENT SEQUENCES AND 805 ARCS Il_AI_E_I_ lJ_LI___a : T08 AGE 60, MEDIUM DENSITY E N T R Y DURATION COSTS VOLUME SIM 5IAIElEl--I_LAiIi__l___A__lA_EA__nL_I__CUI_lELAGL ;_a.t2-I__I4IES I I I I I ! 15 1 1 1 1 1 1 1 F I 8 8 1 2 2 1 1 1 1 F | 5 3 10 5 1 3 40 | 1 2.01 1 T 1 3 1 10 3 1 4 40 1 1 2.01 1 T 1 10 1 1 4 40 1 1 2.01 1 T 1 10 10 I 10 50 I 1 2.01 1 T i THE CORRESPONDING NETWORK HAS 2549 FEASIBLE MANAGEMENT SEQUENCES AND 874 ARCS 

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